Step | Hyp | Ref
| Expression |
1 | | axcc3.2 |
. . 3
⊢ 𝑁 ≈
ω |
2 | | relen 8609 |
. . . 4
⊢ Rel
≈ |
3 | 2 | brrelex1i 5590 |
. . 3
⊢ (𝑁 ≈ ω → 𝑁 ∈ V) |
4 | | mptexg 7015 |
. . 3
⊢ (𝑁 ∈ V → (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V) |
5 | 1, 3, 4 | mp2b 10 |
. 2
⊢ (𝑛 ∈ 𝑁 ↦ 𝐹) ∈ V |
6 | | bren 8614 |
. . . 4
⊢ (𝑁 ≈ ω ↔
∃ℎ ℎ:𝑁–1-1-onto→ω) |
7 | 1, 6 | mpbi 233 |
. . 3
⊢
∃ℎ ℎ:𝑁–1-1-onto→ω |
8 | | axcc2 10016 |
. . . . 5
⊢
∃𝑔(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) |
9 | | f1of 6639 |
. . . . . . . . . . 11
⊢ (ℎ:𝑁–1-1-onto→ω → ℎ:𝑁⟶ω) |
10 | | fnfco 6562 |
. . . . . . . . . . 11
⊢ ((𝑔 Fn ω ∧ ℎ:𝑁⟶ω) → (𝑔 ∘ ℎ) Fn 𝑁) |
11 | 9, 10 | sylan2 596 |
. . . . . . . . . 10
⊢ ((𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → (𝑔 ∘ ℎ) Fn 𝑁) |
12 | 11 | adantlr 715 |
. . . . . . . . 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 1132 |
. . . . . . . 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 5138 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝑛 ∈ 𝑁 ↦ 𝐹) |
15 | 14 | nfeq2 2914 |
. . . . . . . . . 10
⊢
Ⅎ𝑛 𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) |
16 | | nfv 1922 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(𝑔 Fn ω ∧ ∀𝑚 ∈ ω (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ → (𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚))) |
17 | | nfv 1922 |
. . . . . . . . . 10
⊢
Ⅎ𝑛 ℎ:𝑁–1-1-onto→ω |
18 | 15, 16, 17 | nf3an 1909 |
. . . . . . . . 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 6882 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (ℎ‘𝑛) ∈ ω) |
20 | | fveq2 6695 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 = (ℎ‘𝑛) → ((𝑘 ∘ ◡ℎ)‘𝑚) = ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛))) |
21 | 20 | neeq1d 2991 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = (ℎ‘𝑛) → (((𝑘 ∘ ◡ℎ)‘𝑚) ≠ ∅ ↔ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅)) |
22 | | fveq2 6695 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 = (ℎ‘𝑛) → (𝑔‘𝑚) = (𝑔‘(ℎ‘𝑛))) |
23 | 22, 20 | eleq12d 2825 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = (ℎ‘𝑛) → ((𝑔‘𝑚) ∈ ((𝑘 ∘ ◡ℎ)‘𝑚) ↔ (𝑔‘(ℎ‘𝑛)) ∈ ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)))) |
24 | 21, 23 | imbi12d 348 |
. . . . . . . . . . . . . . . . . . 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 3522 |
. . . . . . . . . . . . . . . . . 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 1189 |
. . . . . . . . . . . . . . . 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 6651 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℎ:𝑁–1-1-onto→ω → ◡ℎ:ω–1-1-onto→𝑁) |
29 | | f1of 6639 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (◡ℎ:ω–1-1-onto→𝑁 → ◡ℎ:ω⟶𝑁) |
30 | 28, 29 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (ℎ:𝑁–1-1-onto→ω → ◡ℎ:ω⟶𝑁) |
31 | | fvco3 6788 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((◡ℎ:ω⟶𝑁 ∧ (ℎ‘𝑛) ∈ ω) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
32 | 30, 19, 31 | syl2an2r 685 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
33 | 32 | 3adant1 1132 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = (𝑘‘(◡ℎ‘(ℎ‘𝑛)))) |
34 | | f1ocnvfv1 7065 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (◡ℎ‘(ℎ‘𝑛)) = 𝑛) |
35 | 34 | fveq2d 6699 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘(◡ℎ‘(ℎ‘𝑛))) = (𝑘‘𝑛)) |
36 | 35 | 3adant1 1132 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘(◡ℎ‘(ℎ‘𝑛))) = (𝑘‘𝑛)) |
37 | | fveq1 6694 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (𝑘‘𝑛) = ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛)) |
38 | | axcc3.1 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 𝐹 ∈ V |
39 | | eqid 2736 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ 𝑁 ↦ 𝐹) = (𝑛 ∈ 𝑁 ↦ 𝐹) |
40 | 39 | fvmpt2 6807 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑛 ∈ 𝑁 ∧ 𝐹 ∈ V) → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹) |
41 | 38, 40 | mpan2 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ 𝑁 → ((𝑛 ∈ 𝑁 ↦ 𝐹)‘𝑛) = 𝐹) |
42 | 37, 41 | sylan9eq 2791 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = 𝐹) |
43 | 42 | 3adant2 1133 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → (𝑘‘𝑛) = 𝐹) |
44 | 33, 36, 43 | 3eqtrd 2775 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = 𝐹) |
45 | 44 | 3expa 1120 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = 𝐹) |
46 | 45 | 3adantl2 1169 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) = 𝐹) |
47 | 46 | neeq1d 2991 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → (((𝑘 ∘ ◡ℎ)‘(ℎ‘𝑛)) ≠ ∅ ↔ 𝐹 ≠ ∅)) |
48 | 9 | 3ad2ant3 1137 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) → ℎ:𝑁⟶ω) |
49 | | fvco3 6788 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((ℎ:𝑁⟶ω ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛))) |
50 | 48, 49 | sylan 583 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) ∧ 𝑔 Fn ω ∧ ℎ:𝑁–1-1-onto→ω) ∧ 𝑛 ∈ 𝑁) → ((𝑔 ∘ ℎ)‘𝑛) = (𝑔‘(ℎ‘𝑛))) |
51 | 50 | eleq1d 2815 |
. . . . . . . . . . . . . . . . . 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)))) |
52 | 46 | eleq2d 2816 |
. . . . . . . . . . . . . . . . . 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)) |
53 | 51, 52 | bitr3d 284 |
. . . . . . . . . . . . . . . . 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)) |
54 | 47, 53 | imbi12d 348 |
. . . . . . . . . . . . . . . 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))) |
55 | 27, 54 | sylibd 242 |
. . . . . . . . . . . . . . 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))) |
56 | 55 | ex 416 |
. . . . . . . . . . . . . 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)))) |
57 | 56 | com23 86 |
. . . . . . . . . . . . 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)))) |
58 | 57 | 3exp 1121 |
. . . . . . . . . . . 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)))))) |
59 | 58 | com34 91 |
. . . . . . . . . . 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)))))) |
60 | 59 | imp32 422 |
. . . . . . . . . 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)))) |
61 | 60 | 3impia 1119 |
. . . . . . . . 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))) |
62 | 18, 61 | ralrimi 3127 |
. . . . . . . 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)) |
63 | | vex 3402 |
. . . . . . . . . 10
⊢ 𝑔 ∈ V |
64 | | vex 3402 |
. . . . . . . . . 10
⊢ ℎ ∈ V |
65 | 63, 64 | coex 7686 |
. . . . . . . . 9
⊢ (𝑔 ∘ ℎ) ∈ V |
66 | | fneq1 6448 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓 Fn 𝑁 ↔ (𝑔 ∘ ℎ) Fn 𝑁)) |
67 | | fveq1 6694 |
. . . . . . . . . . . . 13
⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑛) = ((𝑔 ∘ ℎ)‘𝑛)) |
68 | 67 | eleq1d 2815 |
. . . . . . . . . . . 12
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑛) ∈ 𝐹 ↔ ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) |
69 | 68 | imbi2d 344 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹) ↔ (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))) |
70 | 69 | ralbidv 3108 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹) ↔ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹))) |
71 | 66, 70 | anbi12d 634 |
. . . . . . . . 9
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)) ↔ ((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)))) |
72 | 65, 71 | spcev 3511 |
. . . . . . . 8
⊢ (((𝑔 ∘ ℎ) Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → ((𝑔 ∘ ℎ)‘𝑛) ∈ 𝐹)) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))) |
73 | 13, 62, 72 | syl2anc 587 |
. . . . . . 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))) |
74 | 73 | 3exp 1121 |
. . . . . 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))))) |
75 | 74 | exlimdv 1941 |
. . . . 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))))) |
76 | 8, 75 | mpi 20 |
. . . 4
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))) |
77 | 76 | exlimdv 1941 |
. . 3
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → (∃ℎ ℎ:𝑁–1-1-onto→ω → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)))) |
78 | 7, 77 | mpi 20 |
. 2
⊢ (𝑘 = (𝑛 ∈ 𝑁 ↦ 𝐹) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹))) |
79 | 5, 78 | vtocle 3490 |
1
⊢
∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝐹 ≠ ∅ → (𝑓‘𝑛) ∈ 𝐹)) |