Step | Hyp | Ref
| Expression |
1 | | elnn1uz2 11641 |
. . 3
⊢ (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈
(ℤ≥‘2))) |
2 | | vdw.r |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈ Fin) |
3 | | ovex 6577 |
. . . . . . . . . 10
⊢ (1...1)
∈ V |
4 | | elmapg 7757 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Fin ∧ (1...1) ∈
V) → (𝑓 ∈ (𝑅 ↑𝑚
(1...1)) ↔ 𝑓:(1...1)⟶𝑅)) |
5 | 2, 3, 4 | sylancl 693 |
. . . . . . . . 9
⊢ (𝜑 → (𝑓 ∈ (𝑅 ↑𝑚 (1...1)) ↔
𝑓:(1...1)⟶𝑅)) |
6 | 5 | biimpa 500 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) →
𝑓:(1...1)⟶𝑅) |
7 | | 1nn 10908 |
. . . . . . . . . 10
⊢ 1 ∈
ℕ |
8 | | vdwap1 15519 |
. . . . . . . . . 10
⊢ ((1
∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) =
{1}) |
9 | 7, 7, 8 | mp2an 704 |
. . . . . . . . 9
⊢
(1(AP‘1)1) = {1} |
10 | | 1z 11284 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℤ |
11 | | elfz3 12222 |
. . . . . . . . . . . 12
⊢ (1 ∈
ℤ → 1 ∈ (1...1)) |
12 | 10, 11 | mp1i 13 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1)) |
13 | | eqidd 2611 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1)) |
14 | | ffn 5958 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...1)⟶𝑅 → 𝑓 Fn (1...1)) |
15 | 14 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1)) |
16 | | fniniseg 6246 |
. . . . . . . . . . . 12
⊢ (𝑓 Fn (1...1) → (1 ∈
(◡𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧
(𝑓‘1) = (𝑓‘1)))) |
17 | 15, 16 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (1 ∈ (◡𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧
(𝑓‘1) = (𝑓‘1)))) |
18 | 12, 13, 17 | mpbir2and 959 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (◡𝑓 “ {(𝑓‘1)})) |
19 | 18 | snssd 4281 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → {1} ⊆ (◡𝑓 “ {(𝑓‘1)})) |
20 | 9, 19 | syl5eqss 3612 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)})) |
21 | 6, 20 | syldan 486 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) →
(1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)})) |
22 | 21 | ralrimiva 2949 |
. . . . . 6
⊢ (𝜑 → ∀𝑓 ∈ (𝑅 ↑𝑚
(1...1))(1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)})) |
23 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝐾 = 1 → (AP‘𝐾) =
(AP‘1)) |
24 | 23 | oveqd 6566 |
. . . . . . . 8
⊢ (𝐾 = 1 → (1(AP‘𝐾)1) =
(1(AP‘1)1)) |
25 | 24 | sseq1d 3595 |
. . . . . . 7
⊢ (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆
(◡𝑓 “ {(𝑓‘1)}))) |
26 | 25 | ralbidv 2969 |
. . . . . 6
⊢ (𝐾 = 1 → (∀𝑓 ∈ (𝑅 ↑𝑚
(1...1))(1(AP‘𝐾)1)
⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ ∀𝑓 ∈ (𝑅 ↑𝑚
(1...1))(1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))) |
27 | 22, 26 | syl5ibrcom 236 |
. . . . 5
⊢ (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅 ↑𝑚
(1...1))(1(AP‘𝐾)1)
⊆ (◡𝑓 “ {(𝑓‘1)}))) |
28 | | oveq1 6556 |
. . . . . . . . . . . 12
⊢ (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑)) |
29 | 28 | sseq1d 3595 |
. . . . . . . . . . 11
⊢ (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}))) |
30 | | oveq2 6557 |
. . . . . . . . . . . 12
⊢ (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1)) |
31 | 30 | sseq1d 3595 |
. . . . . . . . . . 11
⊢ (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))) |
32 | 29, 31 | rspc2ev 3295 |
. . . . . . . . . 10
⊢ ((1
∈ ℕ ∧ 1 ∈ ℕ ∧ (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})) |
33 | 7, 7, 32 | mp3an12 1406 |
. . . . . . . . 9
⊢
((1(AP‘𝐾)1)
⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})) |
34 | | fvex 6113 |
. . . . . . . . . 10
⊢ (𝑓‘1) ∈
V |
35 | | sneq 4135 |
. . . . . . . . . . . . 13
⊢ (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)}) |
36 | 35 | imaeq2d 5385 |
. . . . . . . . . . . 12
⊢ (𝑐 = (𝑓‘1) → (◡𝑓 “ {𝑐}) = (◡𝑓 “ {(𝑓‘1)})) |
37 | 36 | sseq2d 3596 |
. . . . . . . . . . 11
⊢ (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}))) |
38 | 37 | 2rexbidv 3039 |
. . . . . . . . . 10
⊢ (𝑐 = (𝑓‘1) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}))) |
39 | 34, 38 | spcev 3273 |
. . . . . . . . 9
⊢
(∃𝑎 ∈
ℕ ∃𝑑 ∈
ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐})) |
40 | 33, 39 | syl 17 |
. . . . . . . 8
⊢
((1(AP‘𝐾)1)
⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐})) |
41 | | vdw.k |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈
ℕ0) |
42 | 41 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) →
𝐾 ∈
ℕ0) |
43 | 3, 42, 6 | vdwmc 15520 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) →
(𝐾 MonoAP 𝑓 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}))) |
44 | 40, 43 | syl5ibr 235 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) →
((1(AP‘𝐾)1) ⊆
(◡𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓)) |
45 | 44 | ralimdva 2945 |
. . . . . 6
⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑𝑚
(1...1))(1(AP‘𝐾)1)
⊆ (◡𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))𝐾 MonoAP 𝑓)) |
46 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑛 = 1 → (1...𝑛) = (1...1)) |
47 | 46 | oveq2d 6565 |
. . . . . . . . 9
⊢ (𝑛 = 1 → (𝑅 ↑𝑚 (1...𝑛)) = (𝑅 ↑𝑚
(1...1))) |
48 | 47 | raleqdv 3121 |
. . . . . . . 8
⊢ (𝑛 = 1 → (∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))𝐾 MonoAP 𝑓)) |
49 | 48 | rspcev 3282 |
. . . . . . 7
⊢ ((1
∈ ℕ ∧ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))𝐾 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓) |
50 | 7, 49 | mpan 702 |
. . . . . 6
⊢
(∀𝑓 ∈
(𝑅
↑𝑚 (1...1))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓) |
51 | 45, 50 | syl6 34 |
. . . . 5
⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑𝑚
(1...1))(1(AP‘𝐾)1)
⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
52 | 27, 51 | syld 46 |
. . . 4
⊢ (𝜑 → (𝐾 = 1 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
53 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓)) |
54 | 53 | rexralbidv 3040 |
. . . . . . 7
⊢ (𝑥 = 2 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓)) |
55 | 54 | ralbidv 2969 |
. . . . . 6
⊢ (𝑥 = 2 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓)) |
56 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑓)) |
57 | 56 | rexralbidv 3040 |
. . . . . . 7
⊢ (𝑥 = 𝑘 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓)) |
58 | 57 | ralbidv 2969 |
. . . . . 6
⊢ (𝑥 = 𝑘 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓)) |
59 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓)) |
60 | 59 | rexralbidv 3040 |
. . . . . . 7
⊢ (𝑥 = (𝑘 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓)) |
61 | 60 | ralbidv 2969 |
. . . . . 6
⊢ (𝑥 = (𝑘 + 1) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓)) |
62 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓 ↔ 𝐾 MonoAP 𝑓)) |
63 | 62 | rexralbidv 3040 |
. . . . . . 7
⊢ (𝑥 = 𝐾 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
64 | 63 | ralbidv 2969 |
. . . . . 6
⊢ (𝑥 = 𝐾 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
65 | | hashcl 13009 |
. . . . . . . . . 10
⊢ (𝑟 ∈ Fin →
(#‘𝑟) ∈
ℕ0) |
66 | | nn0p1nn 11209 |
. . . . . . . . . 10
⊢
((#‘𝑟) ∈
ℕ0 → ((#‘𝑟) + 1) ∈ ℕ) |
67 | 65, 66 | syl 17 |
. . . . . . . . 9
⊢ (𝑟 ∈ Fin →
((#‘𝑟) + 1) ∈
ℕ) |
68 | | simpll 786 |
. . . . . . . . . . . 12
⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1))))
∧ ¬ 2 MonoAP 𝑓)
→ 𝑟 ∈
Fin) |
69 | | simplr 788 |
. . . . . . . . . . . . 13
⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1))))
∧ ¬ 2 MonoAP 𝑓)
→ 𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) +
1)))) |
70 | | vex 3176 |
. . . . . . . . . . . . . 14
⊢ 𝑟 ∈ V |
71 | | ovex 6577 |
. . . . . . . . . . . . . 14
⊢
(1...((#‘𝑟) +
1)) ∈ V |
72 | 70, 71 | elmap 7772 |
. . . . . . . . . . . . 13
⊢ (𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1)))
↔ 𝑓:(1...((#‘𝑟) + 1))⟶𝑟) |
73 | 69, 72 | sylib 207 |
. . . . . . . . . . . 12
⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1))))
∧ ¬ 2 MonoAP 𝑓)
→ 𝑓:(1...((#‘𝑟) + 1))⟶𝑟) |
74 | | simpr 476 |
. . . . . . . . . . . 12
⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1))))
∧ ¬ 2 MonoAP 𝑓)
→ ¬ 2 MonoAP 𝑓) |
75 | 68, 73, 74 | vdwlem12 15534 |
. . . . . . . . . . 11
⊢ ¬
((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1))))
∧ ¬ 2 MonoAP 𝑓) |
76 | | iman 439 |
. . . . . . . . . . 11
⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1))))
→ 2 MonoAP 𝑓) ↔
¬ ((𝑟 ∈ Fin ∧
𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1))))
∧ ¬ 2 MonoAP 𝑓)) |
77 | 75, 76 | mpbir 220 |
. . . . . . . . . 10
⊢ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1))))
→ 2 MonoAP 𝑓) |
78 | 77 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1)))2
MonoAP 𝑓) |
79 | | oveq2 6557 |
. . . . . . . . . . . 12
⊢ (𝑛 = ((#‘𝑟) + 1) → (1...𝑛) = (1...((#‘𝑟) + 1))) |
80 | 79 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝑛 = ((#‘𝑟) + 1) → (𝑟 ↑𝑚 (1...𝑛)) = (𝑟 ↑𝑚
(1...((#‘𝑟) +
1)))) |
81 | 80 | raleqdv 3121 |
. . . . . . . . . 10
⊢ (𝑛 = ((#‘𝑟) + 1) → (∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1)))2
MonoAP 𝑓)) |
82 | 81 | rspcev 3282 |
. . . . . . . . 9
⊢
((((#‘𝑟) + 1)
∈ ℕ ∧ ∀𝑓 ∈ (𝑟 ↑𝑚
(1...((#‘𝑟) + 1)))2
MonoAP 𝑓) →
∃𝑛 ∈ ℕ
∀𝑓 ∈ (𝑟 ↑𝑚
(1...𝑛))2 MonoAP 𝑓) |
83 | 67, 78, 82 | syl2anc 691 |
. . . . . . . 8
⊢ (𝑟 ∈ Fin → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓) |
84 | 83 | rgen 2906 |
. . . . . . 7
⊢
∀𝑟 ∈ Fin
∃𝑛 ∈ ℕ
∀𝑓 ∈ (𝑟 ↑𝑚
(1...𝑛))2 MonoAP 𝑓 |
85 | 84 | a1i 11 |
. . . . . 6
⊢ (2 ∈
ℤ → ∀𝑟
∈ Fin ∃𝑛 ∈
ℕ ∀𝑓 ∈
(𝑟
↑𝑚 (1...𝑛))2 MonoAP 𝑓) |
86 | | oveq1 6556 |
. . . . . . . . . . 11
⊢ (𝑟 = 𝑠 → (𝑟 ↑𝑚 (1...𝑛)) = (𝑠 ↑𝑚 (1...𝑛))) |
87 | 86 | raleqdv 3121 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓)) |
88 | 87 | rexbidv 3034 |
. . . . . . . . 9
⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓)) |
89 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚)) |
90 | 89 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → (𝑠 ↑𝑚 (1...𝑛)) = (𝑠 ↑𝑚 (1...𝑚))) |
91 | 90 | raleqdv 3121 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑓)) |
92 | | breq2 4587 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑔)) |
93 | 92 | cbvralv 3147 |
. . . . . . . . . . 11
⊢
(∀𝑓 ∈
(𝑠
↑𝑚 (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) |
94 | 91, 93 | syl6bb 275 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔)) |
95 | 94 | cbvrexv 3148 |
. . . . . . . . 9
⊢
(∃𝑛 ∈
ℕ ∀𝑓 ∈
(𝑠
↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) |
96 | 88, 95 | syl6bb 275 |
. . . . . . . 8
⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔)) |
97 | 96 | cbvralv 3147 |
. . . . . . 7
⊢
(∀𝑟 ∈
Fin ∃𝑛 ∈ ℕ
∀𝑓 ∈ (𝑟 ↑𝑚
(1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) |
98 | | simplr 788 |
. . . . . . . . . 10
⊢ (((𝑘 ∈
(ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin) |
99 | | simpll 786 |
. . . . . . . . . 10
⊢ (((𝑘 ∈
(ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈
(ℤ≥‘2)) |
100 | | simpr 476 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈
(ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) |
101 | 95 | ralbii 2963 |
. . . . . . . . . . 11
⊢
(∀𝑠 ∈
Fin ∃𝑛 ∈ ℕ
∀𝑓 ∈ (𝑠 ↑𝑚
(1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) |
102 | 100, 101 | sylibr 223 |
. . . . . . . . . 10
⊢ (((𝑘 ∈
(ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓) |
103 | 98, 99, 102 | vdwlem11 15533 |
. . . . . . . . 9
⊢ (((𝑘 ∈
(ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓) |
104 | 103 | ex 449 |
. . . . . . . 8
⊢ ((𝑘 ∈
(ℤ≥‘2) ∧ 𝑟 ∈ Fin) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓)) |
105 | 104 | ralrimdva 2952 |
. . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘2) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓)) |
106 | 97, 105 | syl5bi 231 |
. . . . . 6
⊢ (𝑘 ∈
(ℤ≥‘2) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓)) |
107 | 55, 58, 61, 64, 85, 106 | uzind4 11622 |
. . . . 5
⊢ (𝐾 ∈
(ℤ≥‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓) |
108 | | oveq1 6556 |
. . . . . . . 8
⊢ (𝑟 = 𝑅 → (𝑟 ↑𝑚 (1...𝑛)) = (𝑅 ↑𝑚 (1...𝑛))) |
109 | 108 | raleqdv 3121 |
. . . . . . 7
⊢ (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
110 | 109 | rexbidv 3034 |
. . . . . 6
⊢ (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
111 | 110 | rspcv 3278 |
. . . . 5
⊢ (𝑅 ∈ Fin →
(∀𝑟 ∈ Fin
∃𝑛 ∈ ℕ
∀𝑓 ∈ (𝑟 ↑𝑚
(1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
112 | 2, 107, 111 | syl2im 39 |
. . . 4
⊢ (𝜑 → (𝐾 ∈ (ℤ≥‘2)
→ ∃𝑛 ∈
ℕ ∀𝑓 ∈
(𝑅
↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
113 | 52, 112 | jaod 394 |
. . 3
⊢ (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ≥‘2))
→ ∃𝑛 ∈
ℕ ∀𝑓 ∈
(𝑅
↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
114 | 1, 113 | syl5bi 231 |
. 2
⊢ (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
115 | | fveq2 6103 |
. . . . . . 7
⊢ (𝐾 = 0 → (AP‘𝐾) =
(AP‘0)) |
116 | 115 | oveqd 6566 |
. . . . . 6
⊢ (𝐾 = 0 → (1(AP‘𝐾)1) =
(1(AP‘0)1)) |
117 | | vdwap0 15518 |
. . . . . . 7
⊢ ((1
∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) =
∅) |
118 | 7, 7, 117 | mp2an 704 |
. . . . . 6
⊢
(1(AP‘0)1) = ∅ |
119 | 116, 118 | syl6eq 2660 |
. . . . 5
⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = ∅) |
120 | | 0ss 3924 |
. . . . 5
⊢ ∅
⊆ (◡𝑓 “ {(𝑓‘1)}) |
121 | 119, 120 | syl6eqss 3618 |
. . . 4
⊢ (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})) |
122 | 121 | ralrimivw 2950 |
. . 3
⊢ (𝐾 = 0 → ∀𝑓 ∈ (𝑅 ↑𝑚
(1...1))(1(AP‘𝐾)1)
⊆ (◡𝑓 “ {(𝑓‘1)})) |
123 | 122, 51 | syl5 33 |
. 2
⊢ (𝜑 → (𝐾 = 0 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)) |
124 | | elnn0 11171 |
. . 3
⊢ (𝐾 ∈ ℕ0
↔ (𝐾 ∈ ℕ
∨ 𝐾 =
0)) |
125 | 41, 124 | sylib 207 |
. 2
⊢ (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0)) |
126 | 114, 123,
125 | mpjaod 395 |
1
⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓) |