Step  Hyp  Ref
 Expression 
1   fzfi 12633 
. . . . . 6
⊢
(0...𝐾) ∈
Fin 
2   fzfi 12633 
. . . . . 6
⊢
(1...𝑁) ∈
Fin 
3   mapfi 8145 
. . . . . 6
⊢
(((0...𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0...𝐾)
↑_{𝑚} (1...𝑁)) ∈ Fin) 
4  1, 2, 3  mp2an 704 
. . . . 5
⊢
((0...𝐾)
↑_{𝑚} (1...𝑁)) ∈ Fin 
5   fzfi 12633 
. . . . 5
⊢
(0...(𝑁 − 1))
∈ Fin 
6   mapfi 8145 
. . . . 5
⊢
((((0...𝐾)
↑_{𝑚} (1...𝑁)) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → (((0...𝐾) ↑_{𝑚}
(1...𝑁))
↑_{𝑚} (0...(𝑁 − 1))) ∈ Fin) 
7  4, 5, 6  mp2an 704 
. . . 4
⊢
(((0...𝐾)
↑_{𝑚} (1...𝑁)) ↑_{𝑚} (0...(𝑁 − 1))) ∈
Fin 
8  7  a1i 11 
. . 3
⊢ (𝜑 → (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))) ∈
Fin) 
9   2z 11286 
. . . 4
⊢ 2 ∈
ℤ 
10  9  a1i 11 
. . 3
⊢ (𝜑 → 2 ∈
ℤ) 
11   fzofi 12635 
. . . . . . . 8
⊢
(0..^𝐾) ∈
Fin 
12   mapfi 8145 
. . . . . . . 8
⊢
(((0..^𝐾) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((0..^𝐾)
↑_{𝑚} (1...𝑁)) ∈ Fin) 
13  11, 2, 12  mp2an 704 
. . . . . . 7
⊢
((0..^𝐾)
↑_{𝑚} (1...𝑁)) ∈ Fin 
14   mapfi 8145 
. . . . . . . . 9
⊢
(((1...𝑁) ∈ Fin
∧ (1...𝑁) ∈ Fin)
→ ((1...𝑁)
↑_{𝑚} (1...𝑁)) ∈ Fin) 
15  2, 2, 14  mp2an 704 
. . . . . . . 8
⊢
((1...𝑁)
↑_{𝑚} (1...𝑁)) ∈ Fin 
16   f1of 6050 
. . . . . . . . . 10
⊢ (𝑓:(1...𝑁)–11onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁)) 
17  16  ss2abi 3637 
. . . . . . . . 9
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} 
18   ovex 6577 
. . . . . . . . . 10
⊢
(1...𝑁) ∈
V 
19  18, 18  mapval 7756 
. . . . . . . . 9
⊢
((1...𝑁)
↑_{𝑚} (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)} 
20  17, 19  sseqtr4i 3601 
. . . . . . . 8
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)} ⊆ ((1...𝑁) ↑_{𝑚} (1...𝑁)) 
21   ssfi 8065 
. . . . . . . 8
⊢
((((1...𝑁)
↑_{𝑚} (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)} ⊆ ((1...𝑁) ↑_{𝑚} (1...𝑁))) → {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)} ∈ Fin) 
22  15, 20, 21  mp2an 704 
. . . . . . 7
⊢ {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)} ∈ Fin 
23   xpfi 8116 
. . . . . . 7
⊢
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∈ Fin) 
24  13, 22, 23  mp2an 704 
. . . . . 6
⊢
(((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∈ Fin 
25   fzfi 12633 
. . . . . 6
⊢
(0...𝑁) ∈
Fin 
26   xpfi 8116 
. . . . . 6
⊢
(((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin) 
27  24, 25, 26  mp2an 704 
. . . . 5
⊢
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin 
28   rabfi 8070 
. . . . 5
⊢
(((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin) 
29  27, 28  axmp 5 
. . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin 
30   hashcl 13009 
. . . . 5
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈
ℕ_{0}) 
31  30  nn0zd 11356 
. . . 4
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ) 
32  29, 31  mp1i 13 
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ) 
33   dfrex2 2979 
. . . . 5
⊢
(∃𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ↔ ¬ ∀𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) 
34   nfv 1830 
. . . . . 6
⊢
Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 −
1)))) 
35   nfcv 2751 
. . . . . . 7
⊢
Ⅎ𝑡2 
36   nfcv 2751 
. . . . . . 7
⊢
Ⅎ𝑡
∥ 
37   nfcv 2751 
. . . . . . . 8
⊢
Ⅎ𝑡# 
38   nfrab1 3099 
. . . . . . . 8
⊢
Ⅎ𝑡{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} 
39  37, 38  nffv 6110 
. . . . . . 7
⊢
Ⅎ𝑡(#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) 
40  35, 36, 39  nfbr 4629 
. . . . . 6
⊢
Ⅎ𝑡2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) 
41   neq0 3889 
. . . . . . . . . . . 12
⊢ (¬
{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ ↔
∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) 
42   iddvds 14833 
. . . . . . . . . . . . . . . . 17
⊢ (2 ∈
ℤ → 2 ∥ 2) 
43  9, 42  axmp 5 
. . . . . . . . . . . . . . . 16
⊢ 2 ∥
2 
44   vex 3176 
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑠 ∈ V 
45   hashsng 13020 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ V → (#‘{𝑠}) = 1) 
46  44, 45  axmp 5 
. . . . . . . . . . . . . . . . . 18
⊢
(#‘{𝑠}) =
1 
47  46  oveq2i 6560 
. . . . . . . . . . . . . . . . 17
⊢ (1 +
(#‘{𝑠})) = (1 +
1) 
48   df2 10956 
. . . . . . . . . . . . . . . . 17
⊢ 2 = (1 +
1) 
49  47, 48  eqtr4i 2635 
. . . . . . . . . . . . . . . 16
⊢ (1 +
(#‘{𝑠})) =
2 
50  43, 49  breqtrri 4610 
. . . . . . . . . . . . . . 15
⊢ 2 ∥
(1 + (#‘{𝑠})) 
51   rabfi 8070 
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈
Fin) 
52   diffi 8077 
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈ Fin →
({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin) 
53  27, 51, 52  mp2b 10 
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin 
54   snfi 7923 
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑠} ∈ Fin 
55   incom 3767 
. . . . . . . . . . . . . . . . . . . 20
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) 
56   disjdif 3992 
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = ∅ 
57  55, 56  eqtri 2632 
. . . . . . . . . . . . . . . . . . 19
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅ 
58   hashun 13032 
. . . . . . . . . . . . . . . . . . 19
⊢ ((({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin ∧ {𝑠} ∈ Fin ∧ (({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅) → (#‘(({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠}))) 
59  53, 54, 57, 58  mp3an 1416 
. . . . . . . . . . . . . . . . . 18
⊢
(#‘(({𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) 
60   difsnid 4282 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠}) = {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) 
61  60  fveq2d 6107 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (#‘(({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) 
62  59, 61  syl5eqr 2658 
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → ((#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) 
63  62  adantl 481 
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
((#‘({𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) 
64   poimir.0 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℕ) 
65  64  ad3antrrr 762 
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑁 ∈
ℕ) 
66   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑢)) 
67  66  breq2d 4595 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑢))) 
68  67  ifbid 4058 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 𝑢 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑢), 𝑦, (𝑦 + 1))) 
69  68  csbeq1d 3506 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
70   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (1^{st} ‘𝑡) = (1^{st} ‘𝑢)) 
71  70  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑢))) 
72  70  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑡 = 𝑢 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑢))) 
73  72  imaeq1d 5384 
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 𝑢 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑢)) “
(1...𝑗))) 
74  73  xpeq1d 5062 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1})) 
75  72  imaeq1d 5384 
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 𝑢 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑢)) “ ((𝑗 + 1)...𝑁))) 
76  75  xpeq1d 5062 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 𝑢 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) 
77  74, 76  uneq12d 3730 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 𝑢 → ((((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) 
78  71, 77  oveq12d 6567 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 𝑢 → ((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
79  78  csbeq2dv 3944 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2^{nd} ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
80  69, 79  eqtrd 2644 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
81  80  mpteq2dv 4673 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))) 
82   breq1 4586 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 < (2^{nd} ‘𝑢) ↔ 𝑤 < (2^{nd} ‘𝑢))) 
83   id 22 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤) 
84   oveq1 6556 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1)) 
85  82, 83, 84  ifbieq12d 4063 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = 𝑤 → if(𝑦 < (2^{nd} ‘𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2^{nd} ‘𝑢), 𝑤, (𝑤 + 1))) 
86  85  csbeq1d 3506 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2^{nd} ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2^{nd}
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
87   oveq2 6557 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖)) 
88  87  imaeq2d 5385 
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 𝑖 → ((2^{nd}
‘(1^{st} ‘𝑢)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑢)) “
(1...𝑖))) 
89  88  xpeq1d 5062 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 𝑖 → (((2^{nd}
‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1})) 
90   oveq1 6556 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1)) 
91  90  oveq1d 6564 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁)) 
92  91  imaeq2d 5385 
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑗 = 𝑖 → ((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑢)) “ ((𝑖 + 1)...𝑁))) 
93  92  xpeq1d 5062 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑗 = 𝑖 → (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})) 
94  89, 93  uneq12d 3730 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑗 = 𝑖 → ((((2^{nd}
‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))) 
95  94  oveq2d 6565 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 𝑖 → ((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) 
96  95  cbvcsbv 3505 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
⦋if(𝑤
< (2^{nd} ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2^{nd}
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))) 
97  86, 96  syl6eq 2660 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2^{nd} ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2^{nd}
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) 
98  97  cbvmptv 4678 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2^{nd}
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))) 
99  81, 98  syl6eq 2660 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2^{nd}
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))) 
100  99  eqeq2d 2620 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑢 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2^{nd}
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))))) 
101  100  cbvrabv 3172 
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑢 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2^{nd}
‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1^{st}
‘(1^{st} ‘𝑢)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))} 
102   elmapi 7765 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))) →
𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑_{𝑚} (1...𝑁))) 
103  102  ad3antlr 763 
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑_{𝑚} (1...𝑁))) 
104   simpr 476 
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) 
105   simpl 472 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃𝑝 ∈
ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) 
106  105  ralimi 2936 
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) 
107  106  ad2antlr 759 
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) 
108   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑚 → (𝑝‘𝑛) = (𝑝‘𝑚)) 
109  108  neeq1d 2841 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑚) ≠ 0)) 
110  109  rexbidv 3034 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 0)) 
111   fveq1 6102 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = 𝑞 → (𝑝‘𝑚) = (𝑞‘𝑚)) 
112  111  neeq1d 2841 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 0 ↔ (𝑞‘𝑚) ≠ 0)) 
113  112  cbvrexv 3148 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑝 ∈ ran
𝑥(𝑝‘𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) 
114  110, 113  syl6bb 275 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0)) 
115  114  rspccva 3281 
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑛 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) 
116  107, 115  sylan 487 
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0) 
117   simpr 476 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃𝑝 ∈
ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) 
118  117  ralimi 2936 
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) 
119  118  ad2antlr 759 
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) 
120  108  neeq1d 2841 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 𝐾 ↔ (𝑝‘𝑚) ≠ 𝐾)) 
121  120  rexbidv 3034 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 𝐾)) 
122  111  neeq1d 2841 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 𝐾 ↔ (𝑞‘𝑚) ≠ 𝐾)) 
123  122  cbvrexv 3148 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑝 ∈ ran
𝑥(𝑝‘𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) 
124  121, 123  syl6bb 275 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾)) 
125  124  rspccva 3281 
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑛 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) 
126  119, 125  sylan 487 
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾) 
127  65, 101, 103, 104, 116, 126  poimirlem22 32601 
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠) 
128   eldifsn 4260 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) 
129  128  eubii 2480 
. . . . . . . . . . . . . . . . . . 19
⊢
(∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) 
130  53  elexi 3186 
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V 
131   euhash1 13069 
. . . . . . . . . . . . . . . . . . . 20
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V →
((#‘({𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}))) 
132  130, 131  axmp 5 
. . . . . . . . . . . . . . . . . . 19
⊢
((#‘({𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) 
133   dfreu 2903 
. . . . . . . . . . . . . . . . . . 19
⊢
(∃!𝑧 ∈
{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠)) 
134  129, 132,
133  3bitr4ri 292 
. . . . . . . . . . . . . . . . . 18
⊢
(∃!𝑧 ∈
{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ (#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1) 
135  127, 134  sylib 207 
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → (#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1) 
136  135  oveq1d 6564 
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) →
((#‘({𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (1 + (#‘{𝑠}))) 
137  63, 136  eqtr3d 2646 
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (1 + (#‘{𝑠}))) 
138  50, 137  syl5breqr 4621 
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) 
139  138  ex 449 
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) 
140  139  exlimdv 1848 
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) 
141  41, 140  syl5bi 231 
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (¬ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2
∥ (#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) 
142   dvds0 14835 
. . . . . . . . . . . . . 14
⊢ (2 ∈
ℤ → 2 ∥ 0) 
143  9, 142  axmp 5 
. . . . . . . . . . . . 13
⊢ 2 ∥
0 
144   hash0 13019 
. . . . . . . . . . . . 13
⊢
(#‘∅) = 0 
145  143, 144  breqtrri 4610 
. . . . . . . . . . . 12
⊢ 2 ∥
(#‘∅) 
146   fveq2 6103 
. . . . . . . . . . . 12
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ →
(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) =
(#‘∅)) 
147  145, 146  syl5breqr 4621 
. . . . . . . . . . 11
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2
∥ (#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) 
148  141, 147  pm2.61d2 171 
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})) 
149  148  ex 449 
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) 
150  149  adantld 482 
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))) 
151   iba 523 
. . . . . . . . . . 11
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) 
152  151  rabbidv 3164 
. . . . . . . . . 10
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) 
153  152  fveq2d 6107 
. . . . . . . . 9
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) 
154  153  breq2d 4595 
. . . . . . . 8
⊢
(((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ↔ 2 ∥
(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) 
155  150, 154  mpbidi 230 
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) 
156  155  a1d 25 
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})))) 
157  34, 40, 156  rexlimd 3008 
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (∃𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) 
158  33, 157  syl5bir 232 
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (¬ ∀𝑡
∈ ((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))) 
159   simpr 476 
. . . . . . . . 9
⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) 
160  159  con3i 149 
. . . . . . . 8
⊢ (¬
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) 
161  160  ralimi 2936 
. . . . . . 7
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ∀𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) 
162   rabeq0 3911 
. . . . . . 7
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅ ↔ ∀𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) 
163  161, 162  sylibr 223 
. . . . . 6
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅) 
164  163  fveq2d 6107 
. . . . 5
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = (#‘∅)) 
165  145, 164  syl5breqr 4621 
. . . 4
⊢
(∀𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) 
166  158, 165  pm2.61d2 171 
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) 
167  8, 10, 32, 166  fsumdvds 14868 
. 2
⊢ (𝜑 → 2 ∥ Σ𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 −
1)))(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) 
168   rabfi 8070 
. . . . 5
⊢
(((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∈ Fin) 
169  27, 168  axmp 5 
. . . 4
⊢ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∈ Fin 
170   simp1 1054 
. . . . . . 7
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1^{st} ‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶) 
171   sneq 4135 
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘𝑡) = 𝑁 → {(2^{nd} ‘𝑡)} = {𝑁}) 
172  171  difeq2d 3690 
. . . . . . . . . . . 12
⊢
((2^{nd} ‘𝑡) = 𝑁 → ((0...𝑁) ∖ {(2^{nd} ‘𝑡)}) = ((0...𝑁) ∖ {𝑁})) 
173   difun2 4000 
. . . . . . . . . . . . 13
⊢
(((0...(𝑁 −
1)) ∪ {𝑁}) ∖
{𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁}) 
174  64  nnnn0d 11228 
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈
ℕ_{0}) 
175   nn0uz 11598 
. . . . . . . . . . . . . . . . . 18
⊢
ℕ_{0} = (ℤ_{≥}‘0) 
176  174, 175  syl6eleq 2698 
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈
(ℤ_{≥}‘0)) 
177   fzm1 12289 
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ_{≥}‘0) → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) 
178  176, 177  syl 17 
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) 
179   elun 3715 
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁})) 
180   velsn 4141 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁) 
181  180  orbi2i 540 
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) 
182  179, 181  bitri 263 
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) 
183  178, 182  syl6bbr 277 
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}))) 
184  183  eqrdv 2608 
. . . . . . . . . . . . . 14
⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁})) 
185  184  difeq1d 3689 
. . . . . . . . . . . . 13
⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁})) 
186  64  nnzd 11357 
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℤ) 
187   uzid 11578 
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ_{≥}‘𝑁)) 
188   uznfz 12292 
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ_{≥}‘𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1))) 
189  186, 187,
188  3syl 18 
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1))) 
190   disjsn 4192 
. . . . . . . . . . . . . . 15
⊢
(((0...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ ¬ 𝑁 ∈
(0...(𝑁 −
1))) 
191   disj3 3973 
. . . . . . . . . . . . . . 15
⊢
(((0...(𝑁 −
1)) ∩ {𝑁}) = ∅
↔ (0...(𝑁 − 1))
= ((0...(𝑁 − 1))
∖ {𝑁})) 
192  190, 191  bitr3i 265 
. . . . . . . . . . . . . 14
⊢ (¬
𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁})) 
193  189, 192  sylib 207 
. . . . . . . . . . . . 13
⊢ (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁})) 
194  173, 185,
193  3eqtr4a 2670 
. . . . . . . . . . . 12
⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1))) 
195  172, 194  sylan9eqr 2666 
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (2^{nd}
‘𝑡) = 𝑁) → ((0...𝑁) ∖ {(2^{nd} ‘𝑡)}) = (0...(𝑁 − 1))) 
196  195  rexeqdv 3122 
. . . . . . . . . 10
⊢ ((𝜑 ∧ (2^{nd}
‘𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
197  196  biimprd 237 
. . . . . . . . 9
⊢ ((𝜑 ∧ (2^{nd}
‘𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
198  197  ralimdv 2946 
. . . . . . . 8
⊢ ((𝜑 ∧ (2^{nd}
‘𝑡) = 𝑁) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
199  198  expimpd 627 
. . . . . . 7
⊢ (𝜑 → (((2^{nd}
‘𝑡) = 𝑁 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
200  170, 199  sylan2i 685 
. . . . . 6
⊢ (𝜑 → (((2^{nd}
‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
201  200  adantr 480 
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
202  201  ss2rabdv 3646 
. . . 4
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶}) 
203   hashssdif 13061 
. . . 4
⊢ (({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶}) → (#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}))) 
204  169, 202,
203  sylancr 694 
. . 3
⊢ (𝜑 → (#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}))) 
205  64  adantr 480 
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → 𝑁 ∈ ℕ) 
206   poimirlem28.1 
. . . . . . . . . 10
⊢ (𝑝 = ((1^{st} ‘𝑠) ∘_{𝑓} +
((((2^{nd} ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) 
207   poimirlem28.2 
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) 
208  207  adantlr 747 
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) 
209   xp1st 7089 
. . . . . . . . . . . 12
⊢ (𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → (1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)})) 
210   xp1st 7089 
. . . . . . . . . . . 12
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → (1^{st}
‘(1^{st} ‘𝑡)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁))) 
211   elmapi 7765 
. . . . . . . . . . . 12
⊢
((1^{st} ‘(1^{st} ‘𝑡)) ∈ ((0..^𝐾) ↑_{𝑚} (1...𝑁)) → (1^{st}
‘(1^{st} ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) 
212  209, 210,
211  3syl 18 
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → (1^{st}
‘(1^{st} ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) 
213  212  adantl 481 
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (1^{st}
‘(1^{st} ‘𝑡)):(1...𝑁)⟶(0..^𝐾)) 
214   xp2nd 7090 
. . . . . . . . . . . . 13
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → (2^{nd}
‘(1^{st} ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) 
215   fvex 6113 
. . . . . . . . . . . . . 14
⊢
(2^{nd} ‘(1^{st} ‘𝑡)) ∈ V 
216   f1oeq1 6040 
. . . . . . . . . . . . . 14
⊢ (𝑓 = (2^{nd}
‘(1^{st} ‘𝑡)) → (𝑓:(1...𝑁)–11onto→(1...𝑁) ↔ (2^{nd}
‘(1^{st} ‘𝑡)):(1...𝑁)–11onto→(1...𝑁))) 
217  215, 216  elab 3319 
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘(1^{st} ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)} ↔ (2^{nd}
‘(1^{st} ‘𝑡)):(1...𝑁)–11onto→(1...𝑁)) 
218  214, 217  sylib 207 
. . . . . . . . . . . 12
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → (2^{nd}
‘(1^{st} ‘𝑡)):(1...𝑁)–11onto→(1...𝑁)) 
219  209, 218  syl 17 
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → (2^{nd}
‘(1^{st} ‘𝑡)):(1...𝑁)–11onto→(1...𝑁)) 
220  219  adantl 481 
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (2^{nd}
‘(1^{st} ‘𝑡)):(1...𝑁)–11onto→(1...𝑁)) 
221   xp2nd 7090 
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → (2^{nd} ‘𝑡) ∈ (0...𝑁)) 
222  221  adantl 481 
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (2^{nd} ‘𝑡) ∈ (0...𝑁)) 
223  205, 206,
208, 213, 220, 222  poimirlem24 32603 
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋⟨(1^{st}
‘(1^{st} ‘𝑡)), (2^{nd} ‘(1^{st}
‘𝑡))⟩ / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))))) 
224  209  adantl 481 
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)})) 
225   1st2nd2 7096 
. . . . . . . . . . . . . . 15
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → (1^{st} ‘𝑡) = ⟨(1^{st}
‘(1^{st} ‘𝑡)), (2^{nd} ‘(1^{st}
‘𝑡))⟩) 
226  225  csbeq1d 3506 
. . . . . . . . . . . . . 14
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 = ⦋⟨(1^{st}
‘(1^{st} ‘𝑡)), (2^{nd} ‘(1^{st}
‘𝑡))⟩ / 𝑠⦌𝐶) 
227  226  eqeq2d 2620 
. . . . . . . . . . . . 13
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → (𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋⟨(1^{st}
‘(1^{st} ‘𝑡)), (2^{nd} ‘(1^{st}
‘𝑡))⟩ / 𝑠⦌𝐶)) 
228  227  rexbidv 3034 
. . . . . . . . . . . 12
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋⟨(1^{st}
‘(1^{st} ‘𝑡)), (2^{nd} ‘(1^{st}
‘𝑡))⟩ / 𝑠⦌𝐶)) 
229  228  ralbidv 2969 
. . . . . . . . . . 11
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋⟨(1^{st}
‘(1^{st} ‘𝑡)), (2^{nd} ‘(1^{st}
‘𝑡))⟩ / 𝑠⦌𝐶)) 
230  229  anbi1d 737 
. . . . . . . . . 10
⊢
((1^{st} ‘𝑡) ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋⟨(1^{st}
‘(1^{st} ‘𝑡)), (2^{nd} ‘(1^{st}
‘𝑡))⟩ / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))))) 
231  224, 230  syl 17 
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋⟨(1^{st}
‘(1^{st} ‘𝑡)), (2^{nd} ‘(1^{st}
‘𝑡))⟩ / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))))) 
232  223, 231  bitr4d 270 
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))))) 
233   frn 5966 
. . . . . . . . . . . . . . 15
⊢ (𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑_{𝑚} (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) 
234  102, 233  syl 17 
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))) →
ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚}
(1...𝑁))) 
235  234  anim2i 591 
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁)))) 
236   dfss3 3558 
. . . . . . . . . . . . . 14
⊢
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) 
237   vex 3176 
. . . . . . . . . . . . . . . 16
⊢ 𝑛 ∈ V 
238   eqid 2610 
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ ran 𝑥 ↦ 𝐵) = (𝑝 ∈ ran 𝑥 ↦ 𝐵) 
239  238  elrnmpt 5293 
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) 
240  237, 239  axmp 5 
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) 
241  240  ralbii 2963 
. . . . . . . . . . . . . 14
⊢
(∀𝑛 ∈
(0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) 
242  236, 241  sylbb 208 
. . . . . . . . . . . . 13
⊢
((0...(𝑁 − 1))
⊆ ran (𝑝 ∈ ran
𝑥 ↦ 𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) 
243   1eluzge0 11608 
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
(ℤ_{≥}‘0) 
244   fzss1 12251 
. . . . . . . . . . . . . . . . 17
⊢ (1 ∈
(ℤ_{≥}‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))) 
245   ssralv 3629 
. . . . . . . . . . . . . . . . 17
⊢
((1...(𝑁 − 1))
⊆ (0...(𝑁 − 1))
→ (∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) 
246  243, 244,
245  mp2b 10 
. . . . . . . . . . . . . . . 16
⊢
(∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) 
247  64  nncnd 10913 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑁 ∈ ℂ) 
248   npcan1 10334 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) 
249  247, 248  syl 17 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) 
250   peano2zm 11297 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) 
251  186, 250  syl 17 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) 
252   uzid 11578 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) 
253   peano2uz 11617 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑁 − 1) ∈
(ℤ_{≥}‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) 
254  251, 252,
253  3syl 18 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ_{≥}‘(𝑁 − 1))) 
255  249, 254  eqeltrrd 2689 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑁 ∈ (ℤ_{≥}‘(𝑁 − 1))) 
256   fzss2 12252 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑁 ∈
(ℤ_{≥}‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) 
257  255, 256  syl 17 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) 
258  257  sselda 3568 
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁)) 
259  258  adantlr 747 
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁)) 
260   simplr 788 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) 
261   ssel2 3563 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ran
𝑥 ⊆ ((0...𝐾) ↑_{𝑚}
(1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑_{𝑚} (1...𝑁))) 
262   elmapi 7765 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 ∈ ((0...𝐾) ↑_{𝑚} (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾)) 
263  261, 262  syl 17 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ran
𝑥 ⊆ ((0...𝐾) ↑_{𝑚}
(1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾)) 
264  260, 263  sylan 487 
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾)) 
265   poimirlem28.3 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) 
266   elfzelz 12213 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ) 
267  266  zred 11358 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ) 
268  267  ltnrd 10050 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛) 
269   breq1 4586 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 = 𝐵 → (𝑛 < 𝑛 ↔ 𝐵 < 𝑛)) 
270  269  notbid 307 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛)) 
271  268, 270  syl5ibcom 234 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛)) 
272  271  necon2ad 2797 
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) 
273  272  3ad2ant1 1075 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) 
274  273  adantl 481 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵)) 
275  265, 274  mpd 15 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝑛 ≠ 𝐵) 
276  275  3exp2 1277 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵)))) 
277  276  imp31 447 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵)) 
278  277  necon2d 2805 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) 
279  278  adantllr 751 
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) 
280  264, 279  syldan 486 
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0)) 
281  280  reximdva 3000 
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) 
282  259, 281  syldan 486 
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) 
283  282  ralimdva 2945 
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) → (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) 
284  283  imp 444 
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) 
285  246, 284  sylan2 490 
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) 
286  285  biantrurd 528 
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) 
287   nnuz 11599 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ =
(ℤ_{≥}‘1) 
288  64, 287  syl6eleq 2698 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈
(ℤ_{≥}‘1)) 
289   fzm1 12289 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ_{≥}‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) 
290  288, 289  syl 17 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))) 
291   elun 3715 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁})) 
292  180  orbi2i 540 
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) 
293  291, 292  bitri 263 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)) 
294  290, 293  syl6bbr 277 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}))) 
295  294  eqrdv 2608 
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) 
296  295  raleqdv 3121 
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) 
297   ralunb 3756 
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑛 ∈
((1...(𝑁 − 1)) ∪
{𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)) 
298  296, 297  syl6bb 275 
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))) 
299   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑁 → (𝑝‘𝑛) = (𝑝‘𝑁)) 
300  299  neeq1d 2841 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 = 𝑁 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑁) ≠ 0)) 
301  300  rexbidv 3034 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) 
302  301  ralsng 4165 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℕ →
(∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) 
303  64, 302  syl 17 
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) 
304  303  anbi2d 736 
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) 
305  298, 304  bitrd 267 
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) 
306  305  ad2antrr 758 
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))) 
307   0z 11265 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 ∈
ℤ 
308   1z 11284 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℤ 
309   fzshftral 12297 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((0
∈ ℤ ∧ (𝑁
− 1) ∈ ℤ ∧ 1 ∈ ℤ) → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) 
310  307, 308,
309  mp3an13 1407 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑁 − 1) ∈ ℤ
→ (∀𝑛 ∈
(0...(𝑁 −
1))∃𝑝 ∈ ran
𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) 
311  186, 250,
310  3syl 18 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) 
312   0p1e1 11009 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (0 + 1) =
1 
313  312  a1i 11 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (0 + 1) =
1) 
314  313, 249  oveq12d 6567 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁)) 
315  314  raleqdv 3121 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) 
316  311, 315  bitrd 267 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)) 
317   ovex 6577 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 − 1) ∈
V 
318   eqeq1 2614 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵)) 
319  318  rexbidv 3034 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)) 
320  317, 319  sbcie 3437 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
([(𝑚 −
1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵) 
321  320  ralbii 2963 
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
(1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵) 
322   oveq1 6556 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1)) 
323  322  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵)) 
324  323  rexbidv 3034 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)) 
325  324  cbvralv 3147 
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑚 ∈
(1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) 
326  321, 325  bitri 263 
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑚 ∈
(1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) 
327  316, 326  syl6bb 275 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)) 
328  327  biimpa 500 
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) 
329  328  adantlr 747 
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) 
330   poimirlem28.4 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) 
331  330  necomd 2837 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵) 
332  331  3exp2 1277 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵)))) 
333  332  imp31 447 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵)) 
334  333  necon2d 2805 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) 
335  334  adantllr 751 
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) 
336  264, 335  syldan 486 
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾)) 
337  336  reximdva 3000 
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) 
338  337  ralimdva 2945 
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) 
339  338  imp 444 
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) 
340  329, 339  syldan 486 
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) 
341  340  biantrud 527 
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) 
342   r19.26 3046 
. . . . . . . . . . . . . . 15
⊢
(∀𝑛 ∈
(1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) 
343  341, 342  syl6bbr 277 
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) 
344  286, 306,
343  3bitr2d 295 
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑_{𝑚} (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) 
345  235, 242,
344  syl2an 493 
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))) ∧
(0...(𝑁 − 1)) ⊆
ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) 
346  345  pm5.32da 671 
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ (((0...(𝑁 −
1)) ⊆ ran (𝑝 ∈
ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))) 
347  346  anbi2d 736 
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) 
348  347  rexbidva 3031 
. . . . . . . . 9
⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) 
349  348  adantr 480 
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))) 
350  194  rexeqdv 3122 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
351  350  biimpd 218 
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
352  351  ralimdv 2946 
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
353  172  rexeqdv 3122 
. . . . . . . . . . . . . . . . . . 19
⊢
((2^{nd} ‘𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
354  353  ralbidv 2969 
. . . . . . . . . . . . . . . . . 18
⊢
((2^{nd} ‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
355  354  imbi1d 330 
. . . . . . . . . . . . . . . . 17
⊢
((2^{nd} ‘𝑡) = 𝑁 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶))) 
356  352, 355  syl5ibrcom 236 
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((2^{nd}
‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶))) 
357  356  com23 84 
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 → ((2^{nd} ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶))) 
358  357  imp 444 
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶) → ((2^{nd} ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
359  358  adantrd 483 
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶) → (((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶)) 
360  359  pm4.71rd 665 
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶) → (((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))))) 
361   an12 834 
. . . . . . . . . . . . 13
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1^{st} ‘𝑡) / 𝑠⦌𝐶 ∧ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)))) 
362   3anass 1035 
. . . . . . . . . . . . . 14
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1^{st} ‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))) 
363  362  anbi2i 726 
. . . . . . . . . . . . 13
⊢
(((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)))) 
364  361, 363  bitr4i 266 
. . . . . . . . . . . 12
⊢
((∀𝑖 ∈
(0...(𝑁 −
1))∃𝑗 ∈
(0...(𝑁 − 1))𝑖 =
⦋(1^{st} ‘𝑡) / 𝑠⦌𝐶 ∧ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))) 
365  360, 364  syl6bb 275 
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶) → (((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)))) 
366  365  notbid 307 
. . . . . . . . . 10
⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶) → (¬ ((2^{nd}
‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) ↔ ¬ ((2^{nd}
‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)))) 
367  366  pm5.32da 671 
. . . . . . . . 9
⊢ (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))))) 
368  367  adantr 480 
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))))) 
369  232, 349,
368  3bitr3d 297 
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))))) 
370  369  rabbidva 3163 
. . . . . 6
⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)))}) 
371   iunrab 4503 
. . . . . 6
⊢ ∪ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} 
372   difrab 3860 
. . . . . 6
⊢ ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}) = {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)))} 
373  370, 371,
372  3eqtr4g 2669 
. . . . 5
⊢ (𝜑 → ∪ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))})) 
374  373  fveq2d 6107 
. . . 4
⊢ (𝜑 → (#‘∪ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = (#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}))) 
375  27, 28  mp1i 13 
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))))
→ {𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin) 
376   simpl 472 
. . . . . . . . . . . 12
⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))) 
377  376  a1i 11 
. . . . . . . . . . 11
⊢ (𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))))) 
378  377  ss2rabi 3647 
. . . . . . . . . 10
⊢ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} 
379  378  sseli 3564 
. . . . . . . . 9
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) 
380   fveq2 6103 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑠)) 
381  380  breq2d 4595 
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → (𝑦 < (2^{nd} ‘𝑡) ↔ 𝑦 < (2^{nd} ‘𝑠))) 
382  381  ifbid 4058 
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑠 → if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^{nd} ‘𝑠), 𝑦, (𝑦 + 1))) 
383  382  csbeq1d 3506 
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
384   fveq2 6103 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (1^{st} ‘𝑡) = (1^{st} ‘𝑠)) 
385  384  fveq2d 6107 
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑠))) 
386  384  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑠 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑠))) 
387  386  imaeq1d 5384 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑠 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) = ((2^{nd} ‘(1^{st}
‘𝑠)) “
(1...𝑗))) 
388  387  xpeq1d 5062 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^{nd}
‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1})) 
389  386  imaeq1d 5384 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑠 → ((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^{nd} ‘(1^{st}
‘𝑠)) “ ((𝑗 + 1)...𝑁))) 
390  389  xpeq1d 5062 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑠 → (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})) 
391  388, 390  uneq12d 3730 
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑠 → ((((2^{nd}
‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^{nd}
‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))) 
392  385, 391  oveq12d 6567 
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑠 → ((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
393  392  csbeq2dv 3944 
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2^{nd} ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
394  383, 393  eqtrd 2644 
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2^{nd} ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) 
395  394  mpteq2dv 4673 
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑠 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))) 
396  395  eqeq2d 2620 
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))))) 
397   eqcom 2617 
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) 
398  396, 397  syl6bb 275 
. . . . . . . . . . 11
⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)) 
399  398  elrab 3331 
. . . . . . . . . 10
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ↔ (𝑠 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)) 
400  399  simprbi 479 
. . . . . . . . 9
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) 
401  379, 400  syl 17 
. . . . . . . 8
⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥) 
402  401  rgen 2906 
. . . . . . 7
⊢
∀𝑠 ∈
{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 
403  402  rgenw 2908 
. . . . . 6
⊢
∀𝑥 ∈
(((0...𝐾)
↑_{𝑚} (1...𝑁)) ↑_{𝑚} (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 
404   invdisj 4571 
. . . . . 6
⊢
(∀𝑥 ∈
(((0...𝐾)
↑_{𝑚} (1...𝑁)) ↑_{𝑚} (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑠)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 → Disj 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) 
405  403, 404  mp1i 13 
. . . . 5
⊢ (𝜑 → Disj 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) 
406  8, 375, 405  hashiun 14395 
. . . 4
⊢ (𝜑 → (#‘∪ 𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = Σ𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 −
1)))(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) 
407  374, 406  eqtr3d 2646 
. . 3
⊢ (𝜑 → (#‘({𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2^{nd} ‘𝑡)})𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))})) = Σ𝑥 ∈ (((0...𝐾) ↑_{𝑚} (1...𝑁)) ↑_{𝑚}
(0...(𝑁 −
1)))(#‘{𝑡 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^{nd}
‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^{st}
‘(1^{st} ‘𝑡)) ∘_{𝑓} +
((((2^{nd} ‘(1^{st} ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^{nd}
‘(1^{st} ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})) 
408   fo1st 7079 
. . . . . . . . . . . . 13
⊢
1^{st} :V–onto→V 
409   fofun 6029 
. . . . . . . . . . . . 13
⊢
(1^{st} :V–onto→V → Fun 1^{st} ) 
410  408, 409  axmp 5 
. . . . . . . . . . . 12
⊢ Fun
1^{st} 
411   ssv 3588 
. . . . . . . . . . . . 13
⊢ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} ⊆ V 
412   fof 6028 
. . . . . . . . . . . . . . 15
⊢
(1^{st} :V–onto→V → 1^{st}
:V⟶V) 
413  408, 412  axmp 5 
. . . . . . . . . . . . . 14
⊢
1^{st} :V⟶V 
414  413  fdmi 5965 
. . . . . . . . . . . . 13
⊢ dom
1^{st} = V 
415  411, 414  sseqtr4i 3601 
. . . . . . . . . . . 12
⊢ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom
1^{st} 
416   fores 6037 
. . . . . . . . . . . 12
⊢ ((Fun
1^{st} ∧ {𝑡
∈ ((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1^{st} ) →
(1^{st} ↾ {𝑡
∈ ((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}–onto→(1^{st} “ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))})) 
417  410, 415,
416  mp2an 704 
. . . . . . . . . . 11
⊢
(1^{st} ↾ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}–onto→(1^{st} “ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}) 
418   fveq2 6103 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (2^{nd} ‘𝑡) = (2^{nd} ‘𝑥)) 
419  418  eqeq1d 2612 
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ((2^{nd} ‘𝑡) = 𝑁 ↔ (2^{nd} ‘𝑥) = 𝑁)) 
420   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 = 𝑥 → (1^{st} ‘𝑡) = (1^{st} ‘𝑥)) 
421  420  csbeq1d 3506 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑥 → ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶) 
422  421  eqeq2d 2620 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶)) 
423  422  rexbidv 3034 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶)) 
424  423  ralbidv 2969 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶)) 
425  420  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (1^{st}
‘(1^{st} ‘𝑡)) = (1^{st} ‘(1^{st}
‘𝑥))) 
426  425  fveq1d 6105 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = ((1^{st} ‘(1^{st}
‘𝑥))‘𝑁)) 
427  426  eqeq1d 2612 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ↔ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0)) 
428  420  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑥 → (2^{nd}
‘(1^{st} ‘𝑡)) = (2^{nd} ‘(1^{st}
‘𝑥))) 
429  428  fveq1d 6105 
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑥 → ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = ((2^{nd} ‘(1^{st}
‘𝑥))‘𝑁)) 
430  429  eqeq1d 2612 
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑥 → (((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁 ↔ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) 
431  424, 427,
430  3anbi123d 1391 
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑥 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁))) 
432  419, 431  anbi12d 743 
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑥 → (((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)))) 
433  432  rexrab 3337 
. . . . . . . . . . . . . . 15
⊢
(∃𝑥 ∈
{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))(((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠)) 
434   xp1st 7089 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → (1^{st} ‘𝑥) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)})) 
435  434  anim1i 590 
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) → ((1^{st} ‘𝑥) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁))) 
436   eleq1 2676 
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1^{st} ‘𝑥) = 𝑠 → ((1^{st} ‘𝑥) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}))) 
437   csbeq1a 3508 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑠 = (1^{st} ‘𝑥) → 𝐶 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶) 
438  437  eqcoms 2618 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((1^{st} ‘𝑥) = 𝑠 → 𝐶 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶) 
439  438  eqcomd 2616 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((1^{st} ‘𝑥) = 𝑠 → ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 = 𝐶) 
440  439  eqeq2d 2620 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1^{st} ‘𝑥) = 𝑠 → (𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) 
441  440  rexbidv 3034 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1^{st} ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) 
442  441  ralbidv 2969 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1^{st} ‘𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) 
443   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1^{st} ‘𝑥) = 𝑠 → (1^{st}
‘(1^{st} ‘𝑥)) = (1^{st} ‘𝑠)) 
444  443  fveq1d 6105 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1^{st} ‘𝑥) = 𝑠 → ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = ((1^{st} ‘𝑠)‘𝑁)) 
445  444  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1^{st} ‘𝑥) = 𝑠 → (((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ↔ ((1^{st} ‘𝑠)‘𝑁) = 0)) 
446   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1^{st} ‘𝑥) = 𝑠 → (2^{nd}
‘(1^{st} ‘𝑥)) = (2^{nd} ‘𝑠)) 
447  446  fveq1d 6105 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1^{st} ‘𝑥) = 𝑠 → ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = ((2^{nd} ‘𝑠)‘𝑁)) 
448  447  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((1^{st} ‘𝑥) = 𝑠 → (((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁 ↔ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)) 
449  442, 445,
448  3anbi123d 1391 
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1^{st} ‘𝑥) = 𝑠 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁))) 
450  436, 449  anbi12d 743 
. . . . . . . . . . . . . . . . . . . 20
⊢
((1^{st} ‘𝑥) = 𝑠 → (((1^{st} ‘𝑥) ∈ (((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ↔ (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)))) 
451  435, 450  syl5ibcom 234 
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) → ((1^{st} ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)))) 
452  451  adantrl 748 
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∧ ((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁))) → ((1^{st} ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)))) 
453  452  expimpd 627 
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → ((((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)))) 
454  453  rexlimiv 3009 
. . . . . . . . . . . . . . . 16
⊢
(∃𝑥 ∈
((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))(((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁))) 
455   nn0fz0 12306 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ_{0}
↔ 𝑁 ∈ (0...𝑁)) 
456  174, 455  sylib 207 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ (0...𝑁)) 
457   opelxpi 5072 
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) 
458  456, 457  sylan2 490 
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ 𝜑) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) 
459  458  ancoms 468 
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)})) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))) 
460   opelxp2 5075 
. . . . . . . . . . . . . . . . . . . . 21
⊢
(⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁)) 
461   op2ndg 7072 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2^{nd} ‘⟨𝑠, 𝑁⟩) = 𝑁) 
462  461  biantrurd 528 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁) ↔ ((2^{nd} ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)))) 
463   op1stg 7071 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠) 
464   csbeq1a 3508 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑠 = (1^{st}
‘⟨𝑠, 𝑁⟩) → 𝐶 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶) 
465  464  eqcoms 2618 
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠 → 𝐶 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶) 
466  465  eqcomd 2616 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠 → ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 = 𝐶) 
467  463, 466  syl 17 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 = 𝐶) 
468  467  eqeq2d 2620 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶)) 
469  468  rexbidv 3034 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) 
470  469  ralbidv 2969 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶)) 
471  463  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩)) = (1^{st} ‘𝑠)) 
472  471  fveq1d 6105 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((1^{st} ‘𝑠)‘𝑁)) 
473  472  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ↔ ((1^{st} ‘𝑠)‘𝑁) = 0)) 
474  463  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩)) = (2^{nd} ‘𝑠)) 
475  474  fveq1d 6105 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((2^{nd} ‘𝑠)‘𝑁)) 
476  475  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁 ↔ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)) 
477  470, 473,
476  3anbi123d 1391 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁))) 
478  463  biantrud 527 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2^{nd}
‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ↔ (((2^{nd}
‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠))) 
479  462, 477,
478  3bitr3d 297 
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁) ↔ (((2^{nd}
‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠))) 
480  44, 460, 479  sylancr 694 
. . . . . . . . . . . . . . . . . . . 20
⊢
(⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁) ↔ (((2^{nd}
‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠))) 
481  480  biimpa 500 
. . . . . . . . . . . . . . . . . . 19
⊢
((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)) → (((2^{nd}
‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠)) 
482   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (2^{nd} ‘𝑥) = (2^{nd}
‘⟨𝑠, 𝑁⟩)) 
483  482  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((2^{nd} ‘𝑥) = 𝑁 ↔ (2^{nd} ‘⟨𝑠, 𝑁⟩) = 𝑁)) 
484   fveq2 6103 
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (1^{st} ‘𝑥) = (1^{st}
‘⟨𝑠, 𝑁⟩)) 
485  484  csbeq1d 3506 
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶) 
486  485  eqeq2d 2620 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶)) 
487  486  rexbidv 3034 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶)) 
488  487  ralbidv 2969 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶)) 
489  484  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (1^{st}
‘(1^{st} ‘𝑥)) = (1^{st} ‘(1^{st}
‘⟨𝑠, 𝑁⟩))) 
490  489  fveq1d 6105 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = ((1^{st} ‘(1^{st}
‘⟨𝑠, 𝑁⟩))‘𝑁)) 
491  490  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ↔ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0)) 
492  484  fveq2d 6107 
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (2^{nd}
‘(1^{st} ‘𝑥)) = (2^{nd} ‘(1^{st}
‘⟨𝑠, 𝑁⟩))) 
493  492  fveq1d 6105 
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = ((2^{nd} ‘(1^{st}
‘⟨𝑠, 𝑁⟩))‘𝑁)) 
494  493  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁 ↔ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) 
495  488, 491,
494  3anbi123d 1391 
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))) 
496  483, 495  anbi12d 743 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ↔ ((2^{nd}
‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)))) 
497  484  eqeq1d 2612 
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((1^{st} ‘𝑥) = 𝑠 ↔ (1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠)) 
498  496, 497  anbi12d 743 
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((((2^{nd}
‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠) ↔ (((2^{nd} ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠))) 
499  498  rspcev 3282 
. . . . . . . . . . . . . . . . . . 19
⊢
((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∧ (((2^{nd}
‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1^{st} ‘⟨𝑠, 𝑁⟩) = 𝑠)) → ∃𝑥 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))(((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠)) 
500  481, 499  syldan 486 
. . . . . . . . . . . . . . . . . 18
⊢
((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))(((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠)) 
501  459, 500  sylan 487 
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)})) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))(((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠)) 
502  501  expl 646 
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))(((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠))) 
503  454, 502  impbid2 215 
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁))(((2^{nd} ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑥) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑥))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑥))‘𝑁) = 𝑁)) ∧ (1^{st} ‘𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)))) 
504  433, 503  syl5bb 271 
. . . . . . . . . . . . . 14
⊢ (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)))) 
505  504  abbidv 2728 
. . . . . . . . . . . . 13
⊢ (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁))}) 
506   dfimafn 6155 
. . . . . . . . . . . . . . 15
⊢ ((Fun
1^{st} ∧ {𝑡
∈ ((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1^{st} ) →
(1^{st} “ {𝑡
∈ ((((0..^𝐾)
↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑦}) 
507  410, 415,
506  mp2an 704 
. . . . . . . . . . . . . 14
⊢
(1^{st} “ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑦} 
508   nfv 1830 
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(2^{nd} ‘𝑡) = 𝑁 
509   nfcv 2751 
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠(0...(𝑁 − 1)) 
510   nfcsb1v 3515 
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑠⦋(1^{st} ‘𝑡) / 𝑠⦌𝐶 
511  510  nfeq2 2766 
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑠 𝑖 =
⦋(1^{st} ‘𝑡) / 𝑠⦌𝐶 
512  509, 511  nfrex 2990 
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 
513  509, 512  nfral 2929 
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 
514   nfv 1830 
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠((1^{st} ‘(1^{st}
‘𝑡))‘𝑁) = 0 
515   nfv 1830 
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑠((2^{nd} ‘(1^{st}
‘𝑡))‘𝑁) = 𝑁 
516  513, 514,
515  nf3an 1819 
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁) 
517  508, 516  nfan 1816 
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁)) 
518   nfcv 2751 
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑠((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) 
519  517, 518  nfrab 3100 
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} 
520   nfv 1830 
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑠(1^{st} ‘𝑥) = 𝑦 
521  519, 520  nfrex 2990 
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑠∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑦 
522   nfv 1830 
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑦∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑠 
523   eqeq2 2621 
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑠 → ((1^{st} ‘𝑥) = 𝑦 ↔ (1^{st} ‘𝑥) = 𝑠)) 
524  523  rexbidv 3034 
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑠)) 
525  521, 522,
524  cbvab 2733 
. . . . . . . . . . . . . 14
⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑠} 
526  507, 525  eqtri 2632 
. . . . . . . . . . . . 13
⊢
(1^{st} “ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))} (1^{st} ‘𝑥) = 𝑠} 
527   dfrab 2905 
. . . . . . . . . . . . 13
⊢ {𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁))} 
528  505, 526,
527  3eqtr4g 2669 
. . . . . . . . . . . 12
⊢ (𝜑 → (1^{st} “
{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)}) 
529   foeq3 6026 
. . . . . . . . . . . 12
⊢
((1^{st} “ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1^{st} ‘𝑠)‘𝑁) = 0 ∧ ((2^{nd} ‘𝑠)‘𝑁) = 𝑁)} → ((1^{st} ↾ {𝑡 ∈ ((((0..^𝐾) ↑_{𝑚}
(1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
‘(1^{st} ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑_{𝑚} (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–11onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2^{nd} ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1^{st}
‘𝑡) / 𝑠⦌𝐶 ∧ ((1^{st}
‘(1^{st} ‘𝑡))‘𝑁) = 0 ∧ ((2^{nd}
