Theorem poimirlem26 32605

Description: Lemma for poimir 32612 showing an even difference between the number of admissible faces and the number of admissible simplices. Equation (6) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)

Hypotheses

Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem28.1	⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
poimirlem28.2	⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))

Assertion

Ref	Expression
poimirlem26	⊢ (𝜑 → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))

Distinct variable groups:   𝑓,𝑖,𝑗,𝑝,𝑠,𝑡   𝜑,𝑗   𝑗,𝑁   𝜑,𝑖,𝑝,𝑠,𝑡   𝐵,𝑓,𝑖,𝑗,𝑠,𝑡   𝑓,𝐾,𝑖,𝑗,𝑝,𝑠,𝑡   𝑓,𝑁,𝑖,𝑝,𝑠,𝑡   𝐶,𝑖,𝑝,𝑡

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)   𝐶(𝑓,𝑗,𝑠)

Proof of Theorem poimirlem26

Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzofi 12635	. . . . . 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		mapfi 8145	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin)
6	2, 2, 5	mp2an 704	. . . . . 6 ⊢ ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin
7		f1of 6050	. . . . . . . 8 ⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
8	7	ss2abi 3637	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
9		ovex 6577	. . . . . . . 8 ⊢ (1...𝑁) ∈ V
10	9, 9	mapval 7756	. . . . . . 7 ⊢ ((1...𝑁) ↑𝑚 (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
11	8, 10	sseqtr4i 3601	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁))
12		ssfi 8065	. . . . . 6 ⊢ ((((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁))) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
13	6, 11, 12	mp2an 704	. . . . 5 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
14	4, 13	pm3.2i 470	. . . 4 ⊢ (((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
15		xpfi 8116	. . . 4 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
16	14, 15	mp1i 13	. . 3 ⊢ (𝜑 → (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
17		2z 11286	. . . 4 ⊢ 2 ∈ ℤ
18	17	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℤ)
19		snfi 7923	. . . . . . 7 ⊢ {𝑥} ∈ Fin
20		fzfi 12633	. . . . . . . 8 ⊢ (0...𝑁) ∈ Fin
21		rabfi 8070	. . . . . . . 8 ⊢ ((0...𝑁) ∈ Fin → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin)
22	20, 21	ax-mp 5	. . . . . . 7 ⊢ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin
23		xpfi 8116	. . . . . . 7 ⊢ (({𝑥} ∈ Fin ∧ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin) → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin)
24	19, 22, 23	mp2an 704	. . . . . 6 ⊢ ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin
25		hashcl 13009	. . . . . 6 ⊢ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin → (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℕ0)
26	24, 25	ax-mp 5	. . . . 5 ⊢ (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℕ0
27	26	nn0zi 11279	. . . 4 ⊢ (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℤ
28	27	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ∈ ℤ)
29		poimir.0	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
30	29	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 𝑁 ∈ ℕ)
31		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))
32		nfcsb1v 3515	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
33	32	nfeq2 2766	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
34	31, 33	nfim 1813	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
35		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (1...𝑗) = (1...𝑘))
36	35	imaeq2d 5385	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((2nd ‘𝑡) “ (1...𝑗)) = ((2nd ‘𝑡) “ (1...𝑘)))
37	36	xpeq1d 5062	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((2nd ‘𝑡) “ (1...𝑗)) × {1}) = (((2nd ‘𝑡) “ (1...𝑘)) × {1}))
38		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1))
39	38	oveq1d 6564	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → ((𝑗 + 1)...𝑁) = ((𝑘 + 1)...𝑁))
40	39	imaeq2d 5385	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)))
41	40	xpeq1d 5062	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))
42	37, 41	uneq12d 3730	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))
43	42	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))))
44	43	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0})))))
45		csbeq1a 3508	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ⦋𝑡 / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
46	45	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐵 = ⦋𝑡 / 𝑠⦌𝐶 ↔ 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
47	44, 46	imbi12d 333	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶) ↔ (𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)))
48		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑠 𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))
49		nfcsb1v 3515	. . . . . . . . . . . 12 ⊢ Ⅎ𝑠⦋𝑡 / 𝑠⦌𝐶
50	49	nfeq2 2766	. . . . . . . . . . 11 ⊢ Ⅎ𝑠 𝐵 = ⦋𝑡 / 𝑠⦌𝐶
51	48, 50	nfim 1813	. . . . . . . . . 10 ⊢ Ⅎ𝑠(𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶)
52		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → (1st ‘𝑠) = (1st ‘𝑡))
53		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → (2nd ‘𝑠) = (2nd ‘𝑡))
54	53	imaeq1d 5384	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑡) “ (1...𝑗)))
55	54	xpeq1d 5062	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2nd ‘𝑡) “ (1...𝑗)) × {1}))
56	53	imaeq1d 5384	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)))
57	56	xpeq1d 5062	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))
58	55, 57	uneq12d 3730	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))
59	52, 58	oveq12d 6567	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))))
60	59	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0})))))
61		csbeq1a 3508	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → 𝐶 = ⦋𝑡 / 𝑠⦌𝐶)
62	61	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝐵 = 𝐶 ↔ 𝐵 = ⦋𝑡 / 𝑠⦌𝐶))
63	60, 62	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶) ↔ (𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶)))
64		poimirlem28.1	. . . . . . . . . 10 ⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
65	51, 63, 64	chvar 2250	. . . . . . . . 9 ⊢ (𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑡 / 𝑠⦌𝐶)
66	34, 47, 65	chvar 2250	. . . . . . . 8 ⊢ (𝑝 = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑘)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑘 + 1)...𝑁)) × {0}))) → 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
67		poimirlem28.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
68	67	ad4ant14 1285	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
69		xp1st 7089	. . . . . . . . . 10 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑥) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
70		elmapi 7765	. . . . . . . . . 10 ⊢ ((1st ‘𝑥) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾))
71	69, 70	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾))
72	71	ad2antlr 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (1st ‘𝑥):(1...𝑁)⟶(0..^𝐾))
73		xp2nd 7090	. . . . . . . . . 10 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘𝑥) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
74		fvex 6113	. . . . . . . . . . 11 ⊢ (2nd ‘𝑥) ∈ V
75		f1oeq1 6040	. . . . . . . . . . 11 ⊢ (𝑓 = (2nd ‘𝑥) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁)))
76	74, 75	elab 3319	. . . . . . . . . 10 ⊢ ((2nd ‘𝑥) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
77	73, 76	sylib 207	. . . . . . . . 9 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
78	77	ad2antlr 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (2nd ‘𝑥):(1...𝑁)–1-1-onto→(1...𝑁))
79		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑁
80		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑥
81	80, 32	nfcsb 3517	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
82	79, 81	nfne 2882	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
83		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝐶
84	83, 49, 61	cbvcsb 3504	. . . . . . . . . . . . . 14 ⊢ ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑡 / 𝑠⦌𝐶
85	45	csbeq2dv 3944	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ⦋𝑥 / 𝑡⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
86	84, 85	syl5eq 2656	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
87	86	neeq2d 2842	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
88	82, 87	rspc 3276	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
89	88	impcom 445	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
90	89	adantll 746	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
91		1st2nd2 7096	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
92	91	csbeq1d 3506	. . . . . . . . . 10 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
93	92	ad3antlr 763	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
94	90, 93	neeqtrd 2851	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ≠ ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
95	30, 66, 68, 72, 78, 94	poimirlem25 32604	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶}))
96		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = ⦋𝑥 / 𝑠⦌𝐶
97	81	nfeq2 2766	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶
98	86	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
99	96, 97, 98	cbvrex 3144	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶)
100	92	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
101	100	rexbidv 3034	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶))
102	99, 101	syl5rbb 272	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
103	102	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
104		iba 523	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
105	103, 104	sylan9bb 732	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
106	105	rabbidv 3164	. . . . . . . . 9 ⊢ ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → {𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶} = {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
107	106	fveq2d 6107	. . . . . . . 8 ⊢ ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (#‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶}) = (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
108	107	adantll 746	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → (#‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑘 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ / 𝑡⦌⦋𝑘 / 𝑗⦌⦋𝑡 / 𝑠⦌𝐶}) = (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
109	95, 108	breqtrd 4609	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
110	109	ex 449	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
111		dvds0 14835	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∥ 0)
112	17, 111	ax-mp 5	. . . . . . 7 ⊢ 2 ∥ 0
113		hash0 13019	. . . . . . 7 ⊢ (#‘∅) = 0
114	112, 113	breqtrri 4610	. . . . . 6 ⊢ 2 ∥ (#‘∅)
115		simpr 476	. . . . . . . . . 10 ⊢ ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) → ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)
116	115	con3i 149	. . . . . . . . 9 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
117	116	ralrimivw 2950	. . . . . . . 8 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → ∀𝑦 ∈ (0...𝑁) ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
118		rabeq0 3911	. . . . . . . 8 ⊢ ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = ∅ ↔ ∀𝑦 ∈ (0...𝑁) ¬ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
119	117, 118	sylibr 223	. . . . . . 7 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = ∅)
120	119	fveq2d 6107	. . . . . 6 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = (#‘∅))
121	114, 120	syl5breqr 4621	. . . . 5 ⊢ (¬ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
122	110, 121	pm2.61d1 170	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 2 ∥ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
123		hashxp 13081	. . . . . 6 ⊢ (({𝑥} ∈ Fin ∧ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin) → (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = ((#‘{𝑥}) · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
124	19, 22, 123	mp2an 704	. . . . 5 ⊢ (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = ((#‘{𝑥}) · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
125		vex 3176	. . . . . . 7 ⊢ 𝑥 ∈ V
126		hashsng 13020	. . . . . . 7 ⊢ (𝑥 ∈ V → (#‘{𝑥}) = 1)
127	125, 126	ax-mp 5	. . . . . 6 ⊢ (#‘{𝑥}) = 1
128	127	oveq1i 6559	. . . . 5 ⊢ ((#‘{𝑥}) · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (1 · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
129		hashcl 13009	. . . . . . . 8 ⊢ ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∈ Fin → (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ ℕ0)
130	22, 129	ax-mp 5	. . . . . . 7 ⊢ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ ℕ0
131	130	nn0cni 11181	. . . . . 6 ⊢ (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ ℂ
132	131	mulid2i 9922	. . . . 5 ⊢ (1 · (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
133	124, 128, 132	3eqtri 2636	. . . 4 ⊢ (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (#‘{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
134	122, 133	syl6breqr 4625	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → 2 ∥ (#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
135	16, 18, 28, 134	fsumdvds 14868	. 2 ⊢ (𝜑 → 2 ∥ Σ𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
136	4, 13, 15	mp2an 704	. . . . . 6 ⊢ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
137		xpfi 8116	. . . . . 6 ⊢ (((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin)
138	136, 20, 137	mp2an 704	. . . . 5 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
139		rabfi 8070	. . . . 5 ⊢ (((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin)
140	138, 139	ax-mp 5	. . . 4 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin
141	29	nncnd 10913	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
142		npcan1 10334	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
143	141, 142	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
144		nnm1nn0 11211	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
145	29, 144	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
146	145	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
147		uzid 11578	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
148		peano2uz 11617	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
149	146, 147, 148	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
150	143, 149	eqeltrrd 2689	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
151		fzss2 12252	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) → (0...(𝑁 − 1)) ⊆ (0...𝑁))
152		ssralv 3629	. . . . . . . . . 10 ⊢ ((0...(𝑁 − 1)) ⊆ (0...𝑁) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
153	150, 151, 152	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
154	153	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
155		raldifb 3712	. . . . . . . . . . . 12 ⊢ (∀𝑗 ∈ (0...𝑁)(𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
156		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗𝜑
157		nfcsb1v 3515	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
158	157	nfeq2 2766	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
159	156, 158	nfan 1816	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
160		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑖 ∈ (0...(𝑁 − 1))
161	159, 160	nfan 1816	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1)))
162		nnel 2892	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑗 ∉ {(2nd ‘𝑡)} ↔ 𝑗 ∈ {(2nd ‘𝑡)})
163		velsn 4141	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {(2nd ‘𝑡)} ↔ 𝑗 = (2nd ‘𝑡))
164	162, 163	bitri 263	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑗 ∉ {(2nd ‘𝑡)} ↔ 𝑗 = (2nd ‘𝑡))
165		csbeq1a 3508	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑡) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
166	165	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑡) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
167	166	biimparc 503	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ 𝑗 = (2nd ‘𝑡)) → 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
168	29	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ ℝ)
169	168	ltm1d 10835	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
170	145	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
171	170, 168	ltnled 10063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
172	169, 171	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
173		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (0...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
174	172, 173	nsyl 134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
175		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑁 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ 𝑁 ∈ (0...(𝑁 − 1))))
176	175	notbid 307	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑁 → (¬ 𝑖 ∈ (0...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1))))
177	174, 176	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑖 = 𝑁 → ¬ 𝑖 ∈ (0...(𝑁 − 1))))
178	177	con2d 128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑖 ∈ (0...(𝑁 − 1)) → ¬ 𝑖 = 𝑁))
179	178	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ¬ 𝑖 = 𝑁)
180		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑖 = 𝑁 ↔ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
181	180	notbid 307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (¬ 𝑖 = 𝑁 ↔ ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
182	179, 181	syl5ibcom 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
183	167, 182	syl5 33	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ 𝑗 = (2nd ‘𝑡)) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
184	183	expdimp 452	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...(𝑁 − 1))) ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 = (2nd ‘𝑡) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
185	184	an32s 842	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (𝑗 = (2nd ‘𝑡) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
186	164, 185	syl5bi 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
187		idd 24	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
188	186, 187	jad 173	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → ((𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
189	188	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (0...𝑁)) → ((𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
190	161, 189	ralimdaa 2941	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∀𝑗 ∈ (0...𝑁)(𝑗 ∉ {(2nd ‘𝑡)} → ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
191	155, 190	syl5bir 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
192	191	con3d 147	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
193		dfrex2 2979	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∀𝑗 ∈ (0...𝑁) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
194		dfrex2 2979	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∀𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ¬ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
195	192, 193, 194	3imtr4g 284	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑖 ∈ (0...(𝑁 − 1))) → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
196	195	ralimdva 2945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
197	154, 196	syld 46	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
198	197	expimpd 627	. . . . . 6 ⊢ (𝜑 → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
199	198	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
200	199	ss2rabdv 3646	. . . 4 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶})
201		hashssdif 13061	. . . 4 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})))
202	140, 200, 201	sylancr 694	. . 3 ⊢ (𝜑 → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})))
203		xp2nd 7090	. . . . . . . 8 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑡) ∈ (0...𝑁))
204		df-ne 2782	. . . . . . . . . . . 12 ⊢ (𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
205	204	ralbii 2963	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
206		ralnex 2975	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑁) ¬ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
207	205, 206	bitri 263	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
208	29	nnnn0d 11228	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
209		nn0uz 11598	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ0 = (ℤ≥‘0)
210	208, 209	syl6eleq 2698	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
211	143, 210	eqeltrd 2688	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘0))
212		fzsplit2 12237	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
213	211, 150, 212	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
214	143	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
215	29	nnzd 11357	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
216		fzsn 12254	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
217	215, 216	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
218	214, 217	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
219	218	uneq2d 3729	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((0...(𝑁 − 1)) ∪ {𝑁}))
220	213, 219	eqtrd 2644	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
221	220	raleqdv 3121	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
222		difss 3699	. . . . . . . . . . . . . . . . . 18 ⊢ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ⊆ (0...𝑁)
223		ssrexv 3630	. . . . . . . . . . . . . . . . . 18 ⊢ (((0...𝑁) ∖ {(2nd ‘𝑡)}) ⊆ (0...𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
224	222, 223	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
225	224	ralimi 2936	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
226	225	biantrurd 528	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
227		ralunb 3756	. . . . . . . . . . . . . . 15 ⊢ (∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
228	226, 227	syl6rbbr 278	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (∀𝑖 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
229	221, 228	sylan9bb 732	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
230	229	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
231		nn0fz0 12306	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁))
232	208, 231	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
233	232	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 ∈ (0...𝑁))
234		eqeq1 2614	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑁 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
235	234	rexbidv 3034	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑁 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
236	235	rspcva 3280	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
237		nfv 1830	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗(𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁))
238		nfcv 2751	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗(0...(𝑁 − 1))
239		nfre1 2988	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
240	238, 239	nfral 2929	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑗∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
241	237, 240	nfan 1816	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
242		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑁 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
243	242	notbid 307	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (¬ 𝑁 ∈ (0...(𝑁 − 1)) ↔ ¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
244	174, 243	syl5ibcom 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
245	244	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
246		eldifsn 4260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ↔ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd ‘𝑡)))
247		diffi 8077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0...𝑁) ∈ Fin → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin)
248	20, 247	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin
249		ssrab2 3650	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd ‘𝑡)})
250		ssdomg 7887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin → ({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd ‘𝑡)}) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd ‘𝑡)})))
251	248, 249, 250	mp2 9	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd ‘𝑡)})
252		hashdifsn 13063	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((0...𝑁) ∈ Fin ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → (#‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = ((#‘(0...𝑁)) − 1))
253	20, 252	mpan 702	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((2nd ‘𝑡) ∈ (0...𝑁) → (#‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = ((#‘(0...𝑁)) − 1))
254		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → 1 ∈ ℂ)
255	141, 254, 254	addsubd 10292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑁 + 1) − 1) = ((𝑁 − 1) + 1))
256		hashfz0 13079	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 ∈ ℕ0 → (#‘(0...𝑁)) = (𝑁 + 1))
257	208, 256	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (#‘(0...𝑁)) = (𝑁 + 1))
258	257	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((#‘(0...𝑁)) − 1) = ((𝑁 + 1) − 1))
259		hashfz0 13079	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑁 − 1) ∈ ℕ0 → (#‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
260	145, 259	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (#‘(0...(𝑁 − 1))) = ((𝑁 − 1) + 1))
261	255, 258, 260	3eqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((#‘(0...𝑁)) − 1) = (#‘(0...(𝑁 − 1))))
262	253, 261	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → (#‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = (#‘(0...(𝑁 − 1))))
263		fzfi 12633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0...(𝑁 − 1)) ∈ Fin
264		hashen 12997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → ((#‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = (#‘(0...(𝑁 − 1))) ↔ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1))))
265	248, 263, 264	mp2an 704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((#‘((0...𝑁) ∖ {(2nd ‘𝑡)})) = (#‘(0...(𝑁 − 1))) ↔ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)))
266	262, 265	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)))
267		rabfi 8070	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin)
268	248, 267	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin
269		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑖 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
270	269	biimpac 502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
271		rabid 3095	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
272	271	simplbi2com 655	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}))
273	270, 272	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}))
274	273	impancom 455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}))
275	274	ancrd 575	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
276	275	expimpd 627	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → ((𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
277	276	reximdv2 2997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
278	271	simplbi 475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}))
279	274	pm4.71rd 665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
280		df-mpt 4645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) = {⟨𝑘, 𝑖⟩ ∣ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
281		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ Ⅎ𝑘(𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
282		nfrab1 3099	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ Ⅎ𝑗{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
283	282	nfcri 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ Ⅎ𝑗 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
284		nfcsb1v 3515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
285	284	nfeq2 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ Ⅎ𝑗 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
286	283, 285	nfan 1816	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ Ⅎ𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
287		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ 𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}))
288		csbeq1a 3508	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑗 = 𝑘 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
289	288	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑗 = 𝑘 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
290	287, 289	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑗 = 𝑘 → ((𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
291	281, 286, 290	cbvopab1 4655	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} = {⟨𝑘, 𝑖⟩ ∣ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
292	280, 291	eqtr4i 2635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) = {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
293	292	breqi 4589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ 𝑗{⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}𝑖)
294		df-br 4584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑗{⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}𝑖 ↔ ⟨𝑗, 𝑖⟩ ∈ {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})
295		opabid 4907	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (⟨𝑗, 𝑖⟩ ∈ {⟨𝑗, 𝑖⟩ ∣ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
296	293, 294, 295	3bitri 285	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∧ 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
297	279, 296	syl6bbr 277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
298	278, 297	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑖 ∈ (0...(𝑁 − 1)) ∧ 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
299	298	rexbidva 3031	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
300		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ Ⅎ𝑝{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}
301		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ Ⅎ𝑝 𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖
302		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗𝑝
303	282, 284	nfmpt 4674	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
304		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ Ⅎ𝑗𝑖
305	302, 303, 304	nfbr 4629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ Ⅎ𝑗 𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖
306		breq1 4586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑗 = 𝑝 → (𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ 𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
307	282, 300, 301, 305, 306	cbvrexf 3142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑗(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖 ↔ ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖)
308	299, 307	syl6bb 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
309	277, 308	sylibd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑖 ∈ (0...(𝑁 − 1)) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
310	309	ralimia 2934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖)
311		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) = (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
312		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗𝑘
313		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗((0...𝑁) ∖ {(2nd ‘𝑡)})
314	284	nfel1 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))
315	288	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑗 = 𝑘 → (⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
316	312, 313, 314, 315	elrabf 3329	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ (𝑘 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
317	316	simprbi 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
318	311, 317	fmpti 6291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1))
319	310, 318	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
320		dffo4 6283	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1)) ↔ ((𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}⟶(0...(𝑁 − 1)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑝 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}𝑝(𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)𝑖))
321	319, 320	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1)))
322		fodomfi 8124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ∈ Fin ∧ (𝑘 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↦ ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶):{𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}–onto→(0...(𝑁 − 1))) → (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
323	268, 321, 322	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
324		endomtr 7900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((0...𝑁) ∖ {(2nd ‘𝑡)}) ≈ (0...(𝑁 − 1)) ∧ (0...(𝑁 − 1)) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
325	266, 323, 324	syl2an 493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
326		sbth 7965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (({𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≼ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ≼ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))}) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd ‘𝑡)}))
327	251, 325, 326	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd ‘𝑡)}))
328		fisseneq 8056	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((0...𝑁) ∖ {(2nd ‘𝑡)}) ∈ Fin ∧ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ⊆ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∧ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ≈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} = ((0...𝑁) ∖ {(2nd ‘𝑡)}))
329	248, 249, 327, 328	mp3an12i 1420	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} = ((0...𝑁) ∖ {(2nd ‘𝑡)}))
330	329	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} ↔ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})))
331	330	biimpar 501	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → 𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))})
332	288	equcoms 1934	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 = 𝑗 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
333	332	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
334	333	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) ↔ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
335	334, 317	vtoclga 3245	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ {𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)}) ∣ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))} → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
336	331, 335	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
337	246, 336	sylan2br 492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑗 ≠ (2nd ‘𝑡))) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)))
338	337	expr 641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑗 ≠ (2nd ‘𝑡) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1))))
339	338	necon1bd 2800	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (¬ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∈ (0...(𝑁 − 1)) → 𝑗 = (2nd ‘𝑡)))
340	245, 339	syld 46	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑗 = (2nd ‘𝑡)))
341	340	imp 444	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑗 = (2nd ‘𝑡))
342	341, 165	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
343		eqtr 2629	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
344	343	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → (⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
345	344	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
346	342, 345	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ∧ 𝑗 ∈ (0...𝑁)) ∧ 𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
347	346	exp31 628	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (𝑗 ∈ (0...𝑁) → (𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
348	241, 158, 347	rexlimd 3008	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
349	236, 348	syl5 33	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
350	233, 349	mpand 707	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
351	350	pm4.71rd 665	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
352	235	ralsng 4165	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
353	29, 352	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
354	353	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑖 ∈ {𝑁}∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
355	230, 351, 354	3bitr3rd 298	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
356	355	notbid 307	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (¬ ∃𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
357	207, 356	syl5bb 271	. . . . . . . . 9 ⊢ (((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
358	357	pm5.32da 671	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑡) ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))))
359	203, 358	sylan2 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))))
360	359	rabbidva 3163	. . . . . 6 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))})
361		nfv 1830	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑡 = ⟨𝑥, 𝑘⟩
362		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
363		nfrab1 3099	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦{𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}
364	363	nfcri 2745	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}
365	362, 364	nfan 1816	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
366	361, 365	nfan 1816	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
367		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
368		opeq2 4341	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → ⟨𝑥, 𝑘⟩ = ⟨𝑥, 𝑦⟩)
369	368	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → (𝑡 = ⟨𝑥, 𝑘⟩ ↔ 𝑡 = ⟨𝑥, 𝑦⟩))
370		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ 𝑦 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
371		rabid 3095	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
372	370, 371	syl6bb 275	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑦 → (𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ↔ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
373	372	anbi2d 736	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑦 → ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))))
374		3anass 1035	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)) ↔ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
375	373, 374	syl6bbr 277	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑦 → ((𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
376	369, 375	anbi12d 743	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑦 → ((𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ (𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))))
377	366, 367, 376	cbvex 2260	. . . . . . . . . 10 ⊢ (∃𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ ∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
378	377	exbii 1764	. . . . . . . . 9 ⊢ (∃𝑥∃𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) ↔ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
379		eliunxp 5181	. . . . . . . . 9 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ ∃𝑥∃𝑘(𝑡 = ⟨𝑥, 𝑘⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑘 ∈ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
380		elopab 4908	. . . . . . . . 9 ⊢ (𝑡 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))} ↔ ∃𝑥∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))))
381	378, 379, 380	3bitr4i 291	. . . . . . . 8 ⊢ (𝑡 ∈ ∪ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ 𝑡 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))})
382	381	eqriv 2607	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))}
383		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
384	125, 383	op2ndd 7070	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑡) = 𝑦)
385	384	sneqd 4137	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → {(2nd ‘𝑡)} = {𝑦})
386	385	difeq2d 3690	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = ((0...𝑁) ∖ {𝑦}))
387	125, 383	op1std 7069	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑡) = 𝑥)
388	387	csbeq1d 3506	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋𝑥 / 𝑠⦌𝐶)
389	388	eqeq2d 2620	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
390	386, 389	rexeqbidv 3130	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
391	390	ralbidv 2969	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶))
392	388	neeq2d 2842	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
393	392	ralbidv 2969	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))
394	391, 393	anbi12d 743	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑥, 𝑦⟩ → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)))
395	394	rabxp 5078	. . . . . . 7 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑦 ∈ (0...𝑁) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶))}
396	382, 395	eqtr4i 2635	. . . . . 6 ⊢ ∪ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
397		difrab 3860	. . . . . 6 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))}
398	360, 396, 397	3eqtr4g 2669	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}))
399	398	fveq2d 6107	. . . 4 ⊢ (𝜑 → (#‘∪ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})))
400	24	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∈ Fin)
401		inxp 5176	. . . . . . . . . 10 ⊢ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)}))
402		df-ne 2782	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑡 ↔ ¬ 𝑥 = 𝑡)
403		disjsn2 4193	. . . . . . . . . . . . 13 ⊢ (𝑥 ≠ 𝑡 → ({𝑥} ∩ {𝑡}) = ∅)
404	402, 403	sylbir 224	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 = 𝑡 → ({𝑥} ∩ {𝑡}) = ∅)
405	404	xpeq1d 5062	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = 𝑡 → (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = (∅ × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})))
406		0xp 5122	. . . . . . . . . . 11 ⊢ (∅ × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅
407	405, 406	syl6eq 2660	. . . . . . . . . 10 ⊢ (¬ 𝑥 = 𝑡 → (({𝑥} ∩ {𝑡}) × ({𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} ∩ {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)
408	401, 407	syl5eq 2656	. . . . . . . . 9 ⊢ (¬ 𝑥 = 𝑡 → (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)
409	408	orri 390	. . . . . . . 8 ⊢ (𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)
410	409	rgen2w 2909	. . . . . . 7 ⊢ ∀𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅)
411		sneq 4135	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → {𝑥} = {𝑡})
412		csbeq1 3502	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → ⦋𝑥 / 𝑠⦌𝐶 = ⦋𝑡 / 𝑠⦌𝐶)
413	412	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑖 = ⦋𝑡 / 𝑠⦌𝐶))
414	413	rexbidv 3034	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶))
415	414	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶))
416	412	neeq2d 2842	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ 𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶))
417	416	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → (∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶))
418	415, 417	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)))
419	418	rabbidv 3164	. . . . . . . . 9 ⊢ (𝑥 = 𝑡 → {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)} = {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})
420	411, 419	xpeq12d 5064	. . . . . . . 8 ⊢ (𝑥 = 𝑡 → ({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) = ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)}))
421	420	disjor 4567	. . . . . . 7 ⊢ (Disj 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ↔ ∀𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(𝑥 = 𝑡 ∨ (({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}) ∩ ({𝑡} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑡 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑡 / 𝑠⦌𝐶)})) = ∅))
422	410, 421	mpbir 220	. . . . . 6 ⊢ Disj 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})
423	422	a1i 11	. . . . 5 ⊢ (𝜑 → Disj 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)}))
424	16, 400, 423	hashiun 14395	. . . 4 ⊢ (𝜑 → (#‘∪ 𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = Σ𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
425	399, 424	eqtr3d 2646	. . 3 ⊢ (𝜑 → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) = Σ𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})))
426		fo1st 7079	. . . . . . . . . . . 12 ⊢ 1st :V–onto→V
427		fofun 6029	. . . . . . . . . . . 12 ⊢ (1st :V–onto→V → Fun 1st )
428	426, 427	ax-mp 5	. . . . . . . . . . 11 ⊢ Fun 1st
429		ssv 3588	. . . . . . . . . . . 12 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ V
430		fof 6028	. . . . . . . . . . . . . 14 ⊢ (1st :V–onto→V → 1st :V⟶V)
431	426, 430	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1st :V⟶V
432	431	fdmi 5965	. . . . . . . . . . . 12 ⊢ dom 1st = V
433	429, 432	sseqtr4i 3601	. . . . . . . . . . 11 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ dom 1st
434		fores 6037	. . . . . . . . . . 11 ⊢ ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ dom 1st ) → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}))
435	428, 433, 434	mp2an 704	. . . . . . . . . 10 ⊢ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})
436		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → (2nd ‘𝑡) = (2nd ‘𝑥))
437	436	csbeq1d 3506	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
438		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑥 → (1st ‘𝑡) = (1st ‘𝑥))
439	438	csbeq1d 3506	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
440	439	csbeq2dv 3944	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
441	437, 440	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑥 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
442	441	eqeq2d 2620	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑥 → (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
443	439	eqeq2d 2620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
444	443	rexbidv 3034	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
445	444	ralbidv 2969	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑥 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
446	442, 445	anbi12d 743	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑥 → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
447	446	rexrab 3337	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠))
448		xp1st 7089	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
449	448	anim1i 590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
450		eleq1 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑥) = 𝑠 → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})))
451		csbeq1a 3508	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = (1st ‘𝑥) → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
452	451	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1st ‘𝑥) = 𝑠 → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
453	452	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑥) = 𝑠 → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = 𝐶)
454	453	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑥) = 𝑠 → (𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶))
455	454	rexbidv 3034	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
456	455	ralbidv 2969	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑥) = 𝑠 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
457	450, 456	anbi12d 743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑥) = 𝑠 → (((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
458	449, 457	syl5ibcom 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
459	458	adantrl 748	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
460	459	expimpd 627	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
461	460	rexlimiv 3009	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
462		simplr 788	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
463		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0...𝑁) ∈ V
464	463	enref 7874	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...𝑁) ≈ (0...𝑁)
465		phpreu 32563	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((0...𝑁) ∈ Fin ∧ (0...𝑁) ≈ (0...𝑁)) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
466	20, 464, 465	mp2an 704	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
467	466	biimpi 205	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
468		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑁 → (𝑖 = 𝐶 ↔ 𝑁 = 𝐶))
469	468	reubidv 3103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑁 → (∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶))
470	469	rspcva 3280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
471	232, 467, 470	syl2an 493	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
472		riotacl 6525	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶 → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
473	471, 472	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
474	473	adantlr 747	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁))
475		opelxpi 5072	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ (0...𝑁)) → ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
476	462, 474, 475	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
477		riotasbc 6526	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑁 = 𝐶 → [(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶)
478	471, 477	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → [(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶)
479		riotaex 6515	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ V
480		sbceq2g 3942	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) ∈ V → ([(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶 ↔ 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶))
481	479, 480	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ([(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗]𝑁 = 𝐶 ↔ 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)
482	478, 481	sylib 207	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)
483	482	expcom 450	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 → (𝜑 → 𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶))
484	483	imdistanri 723	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
485	484	adantlr 747	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
486		vex 3176	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑠 ∈ V
487	486, 479	op2ndd 7070	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (2nd ‘𝑥) = (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶))
488	487	csbeq1d 3506	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ⦋(2nd ‘𝑥) / 𝑗⦌𝐶 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶)
489		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗𝑠
490		nfriota1 6518	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑗(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)
491	489, 490	nfop 4356	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑗⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩
492	491	nfeq2 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑗 𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩
493	486, 479	op1std 7069	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (1st ‘𝑥) = 𝑠)
494	493	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → 𝑠 = (1st ‘𝑥))
495	494, 451	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
496	492, 495	csbeq2d 3943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ⦋(2nd ‘𝑥) / 𝑗⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
497	488, 496	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
498	497	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
499	495	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (𝑖 = 𝐶 ↔ 𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
500	492, 499	rexbid 3033	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
501	500	ralbidv 2969	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
502	498, 501	anbi12d 743	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ((𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
503	493	biantrud 527	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠)))
504	502, 503	bitr2d 268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ → (((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
505	504	rspcev 3282	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑠, (℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶)⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(℩𝑗 ∈ (0...𝑁)𝑁 = 𝐶) / 𝑗⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠))
506	476, 485, 505	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠))
507	506	expl 646	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠)))
508	461, 507	impbid2 215	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
509	447, 508	syl5bb 271	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)))
510	509	abbidv 2728	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)})
511		dfimafn 6155	. . . . . . . . . . . . . 14 ⊢ ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ dom 1st ) → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦})
512	428, 433, 511	mp2an 704	. . . . . . . . . . . . 13 ⊢ (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦}
513		nfcv 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠(2nd ‘𝑡)
514		nfcsb1v 3515	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠⦋(1st ‘𝑡) / 𝑠⦌𝐶
515	513, 514	nfcsb 3517	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
516	515	nfeq2 2766	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑠 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶
517		nfcv 2751	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠(0...𝑁)
518	514	nfeq2 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
519	517, 518	nfrex 2990	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
520	517, 519	nfral 2929	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑠∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
521	516, 520	nfan 1816	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑠(𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
522		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑠((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
523	521, 522	nfrab 3100	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑠{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}
524		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑠(1st ‘𝑥) = 𝑦
525	523, 524	nfrex 2990	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑠∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦
526		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠
527		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑠 → ((1st ‘𝑥) = 𝑦 ↔ (1st ‘𝑥) = 𝑠))
528	527	rexbidv 3034	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠))
529	525, 526, 528	cbvab 2733	. . . . . . . . . . . . 13 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠}
530	512, 529	eqtri 2632	. . . . . . . . . . . 12 ⊢ (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (1st ‘𝑥) = 𝑠}
531		df-rab 2905	. . . . . . . . . . . 12 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)}
532	510, 530, 531	3eqtr4g 2669	. . . . . . . . . . 11 ⊢ (𝜑 → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
533		foeq3 6026	. . . . . . . . . . 11 ⊢ ((1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
534	532, 533	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
535	435, 534	mpbii 222	. . . . . . . . 9 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
536		fof 6028	. . . . . . . . 9 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
537	535, 536	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
538		fvres 6117	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = (1st ‘𝑥))
539		fvres 6117	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) = (1st ‘𝑦))
540	538, 539	eqeqan12d 2626	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦)))
541	540	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦)))
542	446	elrab 3331	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ↔ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
543		xp2nd 7090	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑥) ∈ (0...𝑁))
544	543	anim1i 590	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) → ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
545	542, 544	sylbi 206	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
546		simpl 472	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
547	546	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶))
548	547	ss2rabi 3647	. . . . . . . . . . . . . . . 16 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶}
549	548	sseli 3564	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶})
550		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑦 → (2nd ‘𝑡) = (2nd ‘𝑦))
551	550	csbeq1d 3506	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶)
552		fveq2 6103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑦 → (1st ‘𝑡) = (1st ‘𝑦))
553	552	csbeq1d 3506	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑦 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑦) / 𝑠⦌𝐶)
554	553	csbeq2dv 3944	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
555	551, 554	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑦 → ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
556	555	eqeq2d 2620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑦 → (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
557	556	elrab 3331	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶} ↔ (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
558		xp2nd 7090	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑦) ∈ (0...𝑁))
559	558	anim1i 590	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶) → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
560	557, 559	sylbi 206	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶} → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
561	549, 560	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
562	545, 561	anim12i 588	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) → (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
563		an4 861	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
564	563	anbi2i 726	. . . . . . . . . . . . . 14 ⊢ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))))
565		anass 679	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
566		ancom 465	. . . . . . . . . . . . . . . . 17 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
567	565, 566	bitr3i 265	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
568	567	anbi1i 727	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
569		anass 679	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))))
570	568, 569	bitri 263	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))))
571		anass 679	. . . . . . . . . . . . . 14 ⊢ (((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))))
572	564, 570, 571	3bitr4i 291	. . . . . . . . . . . . 13 ⊢ ((((2nd ‘𝑥) ∈ (0...𝑁) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)) ∧ ((2nd ‘𝑦) ∈ (0...𝑁) ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)) ↔ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
573	562, 572	sylib 207	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) → ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
574		phpreu 32563	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((0...𝑁) ∈ Fin ∧ (0...𝑁) ≈ (0...𝑁)) → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
575	20, 464, 574	mp2an 704	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
576		reurmo 3138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 → ∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
577	576	ralimi 2936	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑖 ∈ (0...𝑁)∃!𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...𝑁)∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
578	575, 577	sylbi 206	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...𝑁)∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
579		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑁 → (𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
580	579	rmobidv 3108	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑁 → (∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
581	580	rspcva 3280	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (0...𝑁) ∧ ∀𝑖 ∈ (0...𝑁)∃*𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
582	232, 578, 581	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
583		nfv 1830	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘 𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶
584	583	rmo3 3494	. . . . . . . . . . . . . . . . . 18 ⊢ (∃*𝑗 ∈ (0...𝑁)𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑗 ∈ (0...𝑁)∀𝑘 ∈ (0...𝑁)((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → 𝑗 = 𝑘))
585	582, 584	sylib 207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → ∀𝑗 ∈ (0...𝑁)∀𝑘 ∈ (0...𝑁)((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → 𝑗 = 𝑘))
586		nfcsb1v 3515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑗⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶
587	586	nfeq2 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶
588		nfs1v 2425	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑗[𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶
589	587, 588	nfan 1816	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
590		nfv 1830	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑗(2nd ‘𝑥) = 𝑘
591	589, 590	nfim 1813	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑗((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = 𝑘)
592		nfv 1830	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))
593		csbeq1a 3508	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = (2nd ‘𝑥) → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
594	593	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
595	594	anbi1d 737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (2nd ‘𝑥) → ((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
596		eqeq1 2614	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (2nd ‘𝑥) → (𝑗 = 𝑘 ↔ (2nd ‘𝑥) = 𝑘))
597	595, 596	imbi12d 333	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (2nd ‘𝑥) → (((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → 𝑗 = 𝑘) ↔ ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = 𝑘)))
598		sbsbc 3406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
599		vex 3176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑘 ∈ V
600		sbceq2g 3942	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
601	599, 600	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
602	598, 601	bitri 263	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
603		csbeq1 3502	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = (2nd ‘𝑦) → ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)
604	603	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = (2nd ‘𝑦) → (𝑁 = ⦋𝑘 / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
605	602, 604	syl5bb 271	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = (2nd ‘𝑦) → ([𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶))
606	605	anbi2d 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (2nd ‘𝑦) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶)))
607		eqeq2 2621	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (2nd ‘𝑦) → ((2nd ‘𝑥) = 𝑘 ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
608	606, 607	imbi12d 333	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (2nd ‘𝑦) → (((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = 𝑘) ↔ ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
609	591, 592, 597, 608	rspc2 3292	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) → (∀𝑗 ∈ (0...𝑁)∀𝑘 ∈ (0...𝑁)((𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ [𝑘 / 𝑗]𝑁 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → 𝑗 = 𝑘) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
610	585, 609	syl5com 31	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
611	610	impr 647	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)))) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦)))
612		csbeq1 3502	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(1st ‘𝑦) / 𝑠⦌𝐶)
613	612	csbeq2dv 3944	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)
614	613	eqeq2d 2620	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → (𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))
615	614	anbi2d 736	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) ↔ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶)))
616	615	imbi1d 330	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥) = (1st ‘𝑦) → (((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦)) ↔ ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
617	611, 616	syl5ibcom 234	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)))) → ((1st ‘𝑥) = (1st ‘𝑦) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶) → (2nd ‘𝑥) = (2nd ‘𝑦))))
618	617	com23 84	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁)))) → ((𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶) → ((1st ‘𝑥) = (1st ‘𝑦) → (2nd ‘𝑥) = (2nd ‘𝑦))))
619	618	impr 647	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑥) ∈ (0...𝑁) ∧ (2nd ‘𝑦) ∈ (0...𝑁))) ∧ (𝑁 = ⦋(2nd ‘𝑥) / 𝑗⦌⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ 𝑁 = ⦋(2nd ‘𝑦) / 𝑗⦌⦋(1st ‘𝑦) / 𝑠⦌𝐶))) → ((1st ‘𝑥) = (1st ‘𝑦) → (2nd ‘𝑥) = (2nd ‘𝑦)))
620	573, 619	sylan2 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → ((1st ‘𝑥) = (1st ‘𝑦) → (2nd ‘𝑥) = (2nd ‘𝑦)))
621		elrabi 3328	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → 𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
622		elrabi 3328	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} → 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
623		xpopth 7098	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) ↔ 𝑥 = 𝑦))
624	623	biimpd 218	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦))
625	624	expd 451	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((1st ‘𝑥) = (1st ‘𝑦) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦)))
626	621, 622, 625	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) → ((1st ‘𝑥) = (1st ‘𝑦) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦)))
627	626	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → ((1st ‘𝑥) = (1st ‘𝑦) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦)))
628	620, 627	mpdd 42	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦))
629	541, 628	sylbid 229	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) → 𝑥 = 𝑦))
630	629	ralrimivva 2954	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) → 𝑥 = 𝑦))
631		dff13 6416	. . . . . . . 8 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})‘𝑦) → 𝑥 = 𝑦)))
632	537, 630, 631	sylanbrc 695	. . . . . . 7 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
633		df-f1o 5811	. . . . . . 7 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∧ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
634	632, 535, 633	sylanbrc 695	. . . . . 6 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
635		rabfi 8070	. . . . . . . . 9 ⊢ (((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∈ Fin)
636	138, 635	ax-mp 5	. . . . . . . 8 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∈ Fin
637	636	elexi 3186	. . . . . . 7 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∈ V
638	637	f1oen 7862	. . . . . 6 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
639	634, 638	syl 17	. . . . 5 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
640		rabfi 8070	. . . . . . 7 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∈ Fin)
641	136, 640	ax-mp 5	. . . . . 6 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∈ Fin
642		hashen 12997	. . . . . 6 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ∈ Fin) → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
643	636, 641, 642	mp2an 704	. . . . 5 ⊢ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
644	639, 643	sylibr 223	. . . 4 ⊢ (𝜑 → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
645	644	oveq2d 6565	. . 3 ⊢ (𝜑 → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑁 = ⦋(2nd ‘𝑡) / 𝑗⦌⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
646	202, 425, 645	3eqtr3d 2652	. 2 ⊢ (𝜑 → Σ𝑥 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})(#‘({𝑥} × {𝑦 ∈ (0...𝑁) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋𝑥 / 𝑠⦌𝐶 ∧ ∀𝑗 ∈ (0...𝑁)𝑁 ≠ ⦋𝑥 / 𝑠⦌𝐶)})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
647	135, 646	breqtrd 4609	1 ⊢ (𝜑 → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695  [wsb 1867   ∈ wcel 1977  {cab 2596   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ∃*wrmo 2899  {crab 2900  Vcvv 3173  [wsbc 3402  ⦋csb 3499   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  dom cdm 5038   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   ∘_𝑓 cof 6793  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744   ≈ cen 7838   ≼ cdom 7839  Fincfn 7841  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Σcsu 14264   ∥ cdvds 14821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822

This theorem is referenced by:  poimirlem28  32607