Step | Hyp | Ref
| Expression |
1 | | cc3.n |
. . 3
⊢ (𝜑 → 𝑁 ≈ ω) |
2 | | relen 6718 |
. . . 4
⊢ Rel
≈ |
3 | 2 | brrelex1i 4652 |
. . 3
⊢ (𝑁 ≈ ω → 𝑁 ∈ V) |
4 | | mptexg 5718 |
. . 3
⊢ (𝑁 ∈ V → (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V) |
5 | 1, 3, 4 | 3syl 17 |
. 2
⊢ (𝜑 → (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V) |
6 | | bren 6721 |
. . . . . . 7
⊢ (𝑁 ≈ ω ↔
∃ℎ ℎ:𝑁–1-1-onto→ω) |
7 | 1, 6 | sylib 121 |
. . . . . 6
⊢ (𝜑 → ∃ℎ ℎ:𝑁–1-1-onto→ω) |
8 | 7 | adantr 274 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → ∃ℎ ℎ:𝑁–1-1-onto→ω) |
9 | | cc3.cc |
. . . . . . . 8
⊢ (𝜑 →
CCHOICE) |
10 | 9 | ad2antrr 485 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) →
CCHOICE) |
11 | | cc3.f |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑛 ∈ 𝑁 𝐹 ∈ V) |
12 | | eqid 2170 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ 𝑁 ↦ 𝐹) = (𝑛 ∈ 𝑁 ↦ 𝐹) |
13 | 12 | mptfng 5321 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
𝑁 𝐹 ∈ V ↔ (𝑛 ∈ 𝑁 ↦ 𝐹) Fn 𝑁) |
14 | 11, 13 | sylib 121 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑛 ∈ 𝑁 ↦ 𝐹) Fn 𝑁) |
15 | 14 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → (𝑛 ∈ 𝑁 ↦ 𝐹) Fn 𝑁) |
16 | | simpr 109 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) |
17 | 16 | fneq1d 5286 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → (𝑘 Fn 𝑁 ↔ (𝑛 ∈ 𝑁 ↦ 𝐹) Fn 𝑁)) |
18 | 15, 17 | mpbird 166 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → 𝑘 Fn 𝑁) |
19 | 18 | adantr 274 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → 𝑘 Fn 𝑁) |
20 | | f1ocnv 5453 |
. . . . . . . . . 10
⊢ (ℎ:𝑁–1-1-onto→ω → ◡ℎ:ω–1-1-onto→𝑁) |
21 | 20 | adantl 275 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ◡ℎ:ω–1-1-onto→𝑁) |
22 | | f1of 5440 |
. . . . . . . . 9
⊢ (◡ℎ:ω–1-1-onto→𝑁 → ◡ℎ:ω⟶𝑁) |
23 | 21, 22 | syl 14 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ◡ℎ:ω⟶𝑁) |
24 | | fnfco 5370 |
. . . . . . . 8
⊢ ((𝑘 Fn 𝑁 ∧ ◡ℎ:ω⟶𝑁) → (𝑘 ∘ ◡ℎ) Fn ω) |
25 | 19, 23, 24 | syl2anc 409 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → (𝑘 ∘ ◡ℎ) Fn ω) |
26 | 23 | ffvelrnda 5628 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (◡ℎ‘𝑝) ∈ 𝑁) |
27 | | cc3.m |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹) |
28 | 27 | ad3antrrr 489 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹) |
29 | | nfcsb1v 3082 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹 |
30 | 29 | nfcri 2306 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛 𝑤 ∈ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹 |
31 | 30 | nfex 1630 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛∃𝑤 𝑤 ∈ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹 |
32 | | csbeq1a 3058 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (◡ℎ‘𝑝) → 𝐹 = ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹) |
33 | 32 | eleq2d 2240 |
. . . . . . . . . . . 12
⊢ (𝑛 = (◡ℎ‘𝑝) → (𝑤 ∈ 𝐹 ↔ 𝑤 ∈ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹)) |
34 | 33 | exbidv 1818 |
. . . . . . . . . . 11
⊢ (𝑛 = (◡ℎ‘𝑝) → (∃𝑤 𝑤 ∈ 𝐹 ↔ ∃𝑤 𝑤 ∈ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹)) |
35 | 31, 34 | rspc 2828 |
. . . . . . . . . 10
⊢ ((◡ℎ‘𝑝) ∈ 𝑁 → (∀𝑛 ∈ 𝑁 ∃𝑤 𝑤 ∈ 𝐹 → ∃𝑤 𝑤 ∈ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹)) |
36 | 26, 28, 35 | sylc 62 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ∃𝑤 𝑤 ∈ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹) |
37 | | fvco3 5565 |
. . . . . . . . . . . . 13
⊢ ((◡ℎ:ω⟶𝑁 ∧ 𝑝 ∈ ω) → ((𝑘 ∘ ◡ℎ)‘𝑝) = (𝑘‘(◡ℎ‘𝑝))) |
38 | 23, 37 | sylan 281 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ((𝑘 ∘ ◡ℎ)‘𝑝) = (𝑘‘(◡ℎ‘𝑝))) |
39 | | simpllr 529 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) |
40 | 39 | fveq1d 5496 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (𝑘‘(◡ℎ‘𝑝)) = ((𝑛 ∈ 𝑁 ↦ 𝐹)‘(◡ℎ‘𝑝))) |
41 | 11 | ad3antrrr 489 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ∀𝑛 ∈ 𝑁 𝐹 ∈ V) |
42 | 29 | nfel1 2323 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑛⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V |
43 | 32 | eleq1d 2239 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = (◡ℎ‘𝑝) → (𝐹 ∈ V ↔ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V)) |
44 | 42, 43 | rspc 2828 |
. . . . . . . . . . . . . . 15
⊢ ((◡ℎ‘𝑝) ∈ 𝑁 → (∀𝑛 ∈ 𝑁 𝐹 ∈ V → ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V)) |
45 | 26, 41, 44 | sylc 62 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V) |
46 | 12 | fvmpts 5572 |
. . . . . . . . . . . . . 14
⊢ (((◡ℎ‘𝑝) ∈ 𝑁 ∧ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹 ∈ V) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘(◡ℎ‘𝑝)) = ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹) |
47 | 26, 45, 46 | syl2anc 409 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘(◡ℎ‘𝑝)) = ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹) |
48 | 40, 47 | eqtrd 2203 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (𝑘‘(◡ℎ‘𝑝)) = ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹) |
49 | 38, 48 | eqtrd 2203 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ((𝑘 ∘ ◡ℎ)‘𝑝) = ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹) |
50 | 49 | eleq2d 2240 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (𝑤 ∈ ((𝑘 ∘ ◡ℎ)‘𝑝) ↔ 𝑤 ∈ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹)) |
51 | 50 | exbidv 1818 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → (∃𝑤 𝑤 ∈ ((𝑘 ∘ ◡ℎ)‘𝑝) ↔ ∃𝑤 𝑤 ∈ ⦋(◡ℎ‘𝑝) / 𝑛⦌𝐹)) |
52 | 36, 51 | mpbird 166 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑝 ∈ ω) → ∃𝑤 𝑤 ∈ ((𝑘 ∘ ◡ℎ)‘𝑝)) |
53 | 52 | ralrimiva 2543 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ∀𝑝 ∈ ω ∃𝑤 𝑤 ∈ ((𝑘 ∘ ◡ℎ)‘𝑝)) |
54 | 10, 25, 53 | cc2 7216 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ∃𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) |
55 | | simprl 526 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) → 𝑔 Fn ω) |
56 | | f1of 5440 |
. . . . . . . . . 10
⊢ (ℎ:𝑁–1-1-onto→ω → ℎ:𝑁⟶ω) |
57 | 56 | adantl 275 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ℎ:𝑁⟶ω) |
58 | 57 | adantr 274 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) → ℎ:𝑁⟶ω) |
59 | | fnfco 5370 |
. . . . . . . 8
⊢ ((𝑔 Fn ω ∧ ℎ:𝑁⟶ω) → (𝑔 ∘ ℎ) Fn 𝑁) |
60 | 55, 58, 59 | syl2anc 409 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) → (𝑔 ∘ ℎ) Fn 𝑁) |
61 | | nfv 1521 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛𝜑 |
62 | | nfmpt1 4080 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛(𝑛 ∈ 𝑁 ↦ 𝐹) |
63 | 62 | nfeq2 2324 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) |
64 | 61, 63 | nfan 1558 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) |
65 | | nfv 1521 |
. . . . . . . . . 10
⊢
Ⅎ𝑛 ℎ:𝑁–1-1-onto→ω |
66 | 64, 65 | nfan 1558 |
. . . . . . . . 9
⊢
Ⅎ𝑛((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) |
67 | | nfv 1521 |
. . . . . . . . 9
⊢
Ⅎ𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) |
68 | 66, 67 | nfan 1558 |
. . . . . . . 8
⊢
Ⅎ𝑛(((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) |
69 | | fvco3 5565 |
. . . . . . . . . . . . . 14
⊢ ((ℎ:𝑁⟶ω ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛))) |
70 | 58, 69 | sylan 281 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛))) |
71 | | fveq2 5494 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (ℎ‘𝑛) → (𝑔‘𝑚) = (𝑔‘(ℎ‘𝑛))) |
72 | | fveq2 5494 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (ℎ‘𝑛) → ((𝑘 ∘ ◡ℎ)‘𝑚) = ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))) |
73 | 71, 72 | eleq12d 2241 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (ℎ‘𝑛) → ((𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚) ↔ (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)))) |
74 | | simplrr 531 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) |
75 | 58 | ffvelrnda 5628 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (ℎ‘𝑛) ∈ ω) |
76 | 73, 74, 75 | rspcdva 2839 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))) |
77 | 70, 76 | eqeltrd 2247 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))) |
78 | 23 | ad2antrr 485 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ◡ℎ:ω⟶𝑁) |
79 | | fvco3 5565 |
. . . . . . . . . . . . 13
⊢ ((◡ℎ:ω⟶𝑁 ∧ (ℎ‘𝑛) ∈ ω) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
80 | 78, 75, 79 | syl2anc 409 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
81 | 77, 80 | eleqtrd 2249 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) ∈ (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
82 | | simpllr 529 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ℎ:𝑁–1-1-onto→ω) |
83 | | simpr 109 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → 𝑛 ∈ 𝑁) |
84 | | f1ocnvfv1 5753 |
. . . . . . . . . . . . 13
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (◡ℎ‘(ℎ‘𝑛)) = 𝑛) |
85 | 82, 83, 84 | syl2anc 409 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (◡ℎ‘(ℎ‘𝑛)) = 𝑛) |
86 | 85 | fveq2d 5498 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (𝑘‘(◡ℎ‘(ℎ‘𝑛))) = (𝑘‘𝑛)) |
87 | 81, 86 | eleqtrd 2249 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) ∈ (𝑘‘𝑛)) |
88 | 16 | ad3antrrr 489 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) |
89 | 88 | fveq1d 5496 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛)) |
90 | 11 | r19.21bi 2558 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐹 ∈ V) |
91 | 90 | ad5ant15 518 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → 𝐹 ∈ V) |
92 | 12 | fvmpt2 5577 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ 𝑁 ∧ 𝐹 ∈ V) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹) |
93 | 83, 91, 92 | syl2anc 409 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹) |
94 | 89, 93 | eqtrd 2203 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = 𝐹) |
95 | 87, 94 | eleqtrd 2249 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹) |
96 | 95 | ex 114 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) → (𝑛 ∈ 𝑁 → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) |
97 | 68, 96 | ralrimi 2541 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) → ∀𝑛 ∈ 𝑁 ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹) |
98 | | vex 2733 |
. . . . . . . . 9
⊢ 𝑔 ∈ V |
99 | | vex 2733 |
. . . . . . . . 9
⊢ ℎ ∈ V |
100 | 98, 99 | coex 5154 |
. . . . . . . 8
⊢ (𝑔 ∘ ℎ) ∈ V |
101 | | fneq1 5284 |
. . . . . . . . 9
⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓 Fn 𝑁 ↔ (𝑔 ∘ ℎ) Fn 𝑁)) |
102 | | fveq1 5493 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑛) = ((𝑔 ∘ ℎ)‘𝑛)) |
103 | 102 | eleq1d 2239 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑛) ∈ 𝐹 ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) |
104 | 103 | ralbidv 2470 |
. . . . . . . . 9
⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹 ↔ ∀𝑛 ∈ 𝑁 ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) |
105 | 101, 104 | anbi12d 470 |
. . . . . . . 8
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹) ↔ ((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))) |
106 | 100, 105 | spcev 2825 |
. . . . . . 7
⊢ (((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹)) |
107 | 60, 97, 106 | syl2anc 409 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹)) |
108 | 54, 107 | exlimddv 1891 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) ∧ ℎ:𝑁–1-1-onto→ω) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹)) |
109 | 8, 108 | exlimddv 1891 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹)) |
110 | 109 | expcom 115 |
. . 3
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))) |
111 | 110 | vtocleg 2801 |
. 2
⊢ ((𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V → (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹))) |
112 | 5, 111 | mpcom 36 |
1
⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐹)) |