Step | Hyp | Ref
| Expression |
1 | | axcc3.2 |
. . 3
⊢ 𝑁 ≈
ω |
2 | | relen 7846 |
. . . 4
⊢ Rel
≈ |
3 | 2 | brrelexi 5082 |
. . 3
⊢ (𝑁 ≈ ω → 𝑁 ∈ V) |
4 | | mptexg 6389 |
. . 3
⊢ (𝑁 ∈ V → (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V) |
5 | 1, 3, 4 | mp2b 10 |
. 2
⊢ (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V |
6 | | bren 7850 |
. . . 4
⊢ (𝑁 ≈ ω ↔
∃ℎ ℎ:𝑁–1-1-onto→ω) |
7 | 1, 6 | mpbi 219 |
. . 3
⊢
∃ℎ ℎ:𝑁–1-1-onto→ω |
8 | | axcc2 9142 |
. . . . 5
⊢
∃𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) |
9 | | f1of 6050 |
. . . . . . . . . . 11
⊢ (ℎ:𝑁–1-1-onto→ω → ℎ:𝑁⟶ω) |
10 | | fnfco 5982 |
. . . . . . . . . . 11
⊢ ((𝑔 Fn ω ∧ ℎ:𝑁⟶ω) → (𝑔 ∘ ℎ) Fn 𝑁) |
11 | 9, 10 | sylan2 490 |
. . . . . . . . . 10
⊢ ((𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → (𝑔 ∘ ℎ) Fn 𝑁) |
12 | 11 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → (𝑔 ∘ ℎ) Fn 𝑁) |
13 | 12 | 3adant1 1072 |
. . . . . . . 8
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → (𝑔 ∘ ℎ) Fn 𝑁) |
14 | | nfmpt1 4675 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝑛 ∈ 𝑁 ↦ 𝐹) |
15 | 14 | nfeq2 2766 |
. . . . . . . . . 10
⊢
Ⅎ𝑛 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) |
16 | | nfv 1830 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) |
17 | | nfv 1830 |
. . . . . . . . . 10
⊢
Ⅎ𝑛 ℎ:𝑁–1-1-onto→ω |
18 | 15, 16, 17 | nf3an 1819 |
. . . . . . . . 9
⊢
Ⅎ𝑛(𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) |
19 | 9 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (ℎ‘𝑛) ∈ ω) |
20 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 = (ℎ‘𝑛) → ((𝑘 ∘ ◡ℎ)‘𝑚) = ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))) |
21 | 20 | neeq1d 2841 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = (ℎ‘𝑛) → (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ ↔ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅)) |
22 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 = (ℎ‘𝑛) → (𝑔‘𝑚) = (𝑔‘(ℎ‘𝑛))) |
23 | 22, 20 | eleq12d 2682 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = (ℎ‘𝑛) → ((𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚) ↔ (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)))) |
24 | 21, 23 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 = (ℎ‘𝑛) → ((((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) ↔ (((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))))) |
25 | 24 | rspcv 3278 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℎ‘𝑛) ∈ ω → (∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) → (((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))))) |
26 | 19, 25 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) → (((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))))) |
27 | 26 | 3ad2antl3 1218 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) → (((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))))) |
28 | | f1ocnv 6062 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (ℎ:𝑁–1-1-onto→ω → ◡ℎ:ω–1-1-onto→𝑁) |
29 | | f1of 6050 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (◡ℎ:ω–1-1-onto→𝑁 → ◡ℎ:ω⟶𝑁) |
30 | 28, 29 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℎ:𝑁–1-1-onto→ω → ◡ℎ:ω⟶𝑁) |
31 | | fvco3 6185 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((◡ℎ:ω⟶𝑁 ∧ (ℎ‘𝑛) ∈ ω) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
32 | 30, 31 | sylan 487 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ (ℎ‘𝑛) ∈ ω) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
33 | 19, 32 | syldan 486 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
34 | 33 | 3adant1 1072 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
35 | | f1ocnvfv1 6432 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (◡ℎ‘(ℎ‘𝑛)) = 𝑛) |
36 | 35 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘(◡ℎ‘(ℎ‘𝑛))) = (𝑘‘𝑛)) |
37 | 36 | 3adant1 1072 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘(◡ℎ‘(ℎ‘𝑛))) = (𝑘‘𝑛)) |
38 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝑘‘𝑛) = ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛)) |
39 | | axcc3.1 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 𝐹 ∈ V |
40 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ 𝑁 ↦ 𝐹) = (𝑛 ∈ 𝑁 ↦ 𝐹) |
41 | 40 | fvmpt2 6200 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑛 ∈ 𝑁 ∧ 𝐹 ∈ V) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹) |
42 | 39, 41 | mpan2 703 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ 𝑁 → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹) |
43 | 38, 42 | sylan9eq 2664 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = 𝐹) |
44 | 43 | 3adant2 1073 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = 𝐹) |
45 | 34, 37, 44 | 3eqtrd 2648 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = 𝐹) |
46 | 45 | 3expa 1257 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = 𝐹) |
47 | 46 | 3adantl2 1211 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = 𝐹) |
48 | 47 | neeq1d 2841 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ ↔ 𝐹 ≠ ∅)) |
49 | 9 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → ℎ:𝑁⟶ω) |
50 | | fvco3 6185 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((ℎ:𝑁⟶ω ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛))) |
51 | 49, 50 | sylan 487 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛))) |
52 | 51 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (((𝑔 ∘ ℎ)‘𝑛) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ↔ (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)))) |
53 | 47 | eleq2d 2673 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (((𝑔 ∘ ℎ)‘𝑛) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) |
54 | 52, 53 | bitr3d 269 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) |
55 | 48, 54 | imbi12d 333 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ → (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))) ↔ (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))) |
56 | 27, 55 | sylibd 228 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))) |
57 | 56 | ex 449 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → (𝑛 ∈ 𝑁 → (∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))) |
58 | 57 | com23 84 |
. . . . . . . . . . . . 13
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → (∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))) |
59 | 58 | 3exp 1256 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝑔 Fn ω → (ℎ:𝑁–1-1-onto→ω → (∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))))) |
60 | 59 | com34 89 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝑔 Fn ω → (∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)) → (ℎ:𝑁–1-1-onto→ω → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))))) |
61 | 60 | imp32 448 |
. . . . . . . . . 10
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚)))) → (ℎ:𝑁–1-1-onto→ω → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))) |
62 | 61 | 3impia 1253 |
. . . . . . . . 9
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → (𝑛 ∈ 𝑁 → (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))) |
63 | 18, 62 | ralrimi 2940 |
. . . . . . . 8
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) |
64 | | vex 3176 |
. . . . . . . . . 10
⊢ 𝑔 ∈ V |
65 | | vex 3176 |
. . . . . . . . . 10
⊢ ℎ ∈ V |
66 | 64, 65 | coex 7011 |
. . . . . . . . 9
⊢ (𝑔 ∘ ℎ) ∈ V |
67 | | fneq1 5893 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓 Fn 𝑁 ↔ (𝑔 ∘ ℎ) Fn 𝑁)) |
68 | | fveq1 6102 |
. . . . . . . . . . . . 13
⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑛) = ((𝑔 ∘ ℎ)‘𝑛)) |
69 | 68 | eleq1d 2672 |
. . . . . . . . . . . 12
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑛) ∈ 𝐹 ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) |
70 | 69 | imbi2d 329 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹) ↔ (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))) |
71 | 70 | ralbidv 2969 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹) ↔ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))) |
72 | 67, 71 | anbi12d 743 |
. . . . . . . . 9
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)) ↔ ((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))) |
73 | 66, 72 | spcev 3273 |
. . . . . . . 8
⊢ (((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))) |
74 | 13, 63, 73 | syl2anc 691 |
. . . . . . 7
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ (𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) ∧ ℎ:𝑁–1-1-onto→ω) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))) |
75 | 74 | 3exp 1256 |
. . . . . 6
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → ((𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) → (ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))))) |
76 | 75 | exlimdv 1848 |
. . . . 5
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (∃𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) → (ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))))) |
77 | 8, 76 | mpi 20 |
. . . 4
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))) |
78 | 77 | exlimdv 1848 |
. . 3
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (∃ℎ ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))) |
79 | 7, 78 | mpi 20 |
. 2
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))) |
80 | 5, 79 | vtocle 3255 |
1
⊢
∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)) |