Step | Hyp | Ref
| Expression |
1 | | bren 6681 |
. . . . . . 7
⊢ (𝐴 ≈ 𝐵 ↔ ∃ℎ ℎ:𝐴–1-1-onto→𝐵) |
2 | 1 | biimpi 119 |
. . . . . 6
⊢ (𝐴 ≈ 𝐵 → ∃ℎ ℎ:𝐴–1-1-onto→𝐵) |
3 | 2 | ad2antrr 480 |
. . . . 5
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) → ∃ℎ ℎ:𝐴–1-1-onto→𝐵) |
4 | | fveq1 5460 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑔 ∘ ℎ) → (𝑓‘𝑥) = ((𝑔 ∘ ℎ)‘𝑥)) |
5 | 4 | eqeq1d 2163 |
. . . . . . . . 9
⊢ (𝑓 = (𝑔 ∘ ℎ) → ((𝑓‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘𝑥) = 1o)) |
6 | 5 | ralbidv 2454 |
. . . . . . . 8
⊢ (𝑓 = (𝑔 ∘ ℎ) → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)) |
7 | 6 | dcbid 824 |
. . . . . . 7
⊢ (𝑓 = (𝑔 ∘ ℎ) → (DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ DECID
∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o)) |
8 | | iswomnimap 7088 |
. . . . . . . . 9
⊢ (𝐴 ∈ WOmni → (𝐴 ∈ WOmni ↔
∀𝑓 ∈
(2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) |
9 | 8 | ibi 175 |
. . . . . . . 8
⊢ (𝐴 ∈ WOmni →
∀𝑓 ∈
(2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) |
10 | 9 | ad3antlr 485 |
. . . . . . 7
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ∀𝑓 ∈ (2o
↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) |
11 | | simpr 109 |
. . . . . . . . . . 11
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) → 𝑔 ∈ (2o
↑𝑚 𝐵)) |
12 | | 2onn 6457 |
. . . . . . . . . . . . 13
⊢
2o ∈ ω |
13 | | relen 6678 |
. . . . . . . . . . . . . 14
⊢ Rel
≈ |
14 | 13 | brrelex2i 4623 |
. . . . . . . . . . . . 13
⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
15 | | elmapg 6595 |
. . . . . . . . . . . . 13
⊢
((2o ∈ ω ∧ 𝐵 ∈ V) → (𝑔 ∈ (2o
↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o)) |
16 | 12, 14, 15 | sylancr 411 |
. . . . . . . . . . . 12
⊢ (𝐴 ≈ 𝐵 → (𝑔 ∈ (2o
↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o)) |
17 | 16 | ad2antrr 480 |
. . . . . . . . . . 11
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) → (𝑔 ∈ (2o
↑𝑚 𝐵) ↔ 𝑔:𝐵⟶2o)) |
18 | 11, 17 | mpbid 146 |
. . . . . . . . . 10
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) → 𝑔:𝐵⟶2o) |
19 | 18 | adantr 274 |
. . . . . . . . 9
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝑔:𝐵⟶2o) |
20 | | f1of 5407 |
. . . . . . . . . 10
⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ:𝐴⟶𝐵) |
21 | 20 | adantl 275 |
. . . . . . . . 9
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ℎ:𝐴⟶𝐵) |
22 | | fco 5328 |
. . . . . . . . 9
⊢ ((𝑔:𝐵⟶2o ∧ ℎ:𝐴⟶𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o) |
23 | 19, 21, 22 | syl2anc 409 |
. . . . . . . 8
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ):𝐴⟶2o) |
24 | | simpllr 524 |
. . . . . . . . 9
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → 𝐴 ∈ WOmni) |
25 | | elmapg 6595 |
. . . . . . . . 9
⊢
((2o ∈ ω ∧ 𝐴 ∈ WOmni) → ((𝑔 ∘ ℎ) ∈ (2o
↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o)) |
26 | 12, 24, 25 | sylancr 411 |
. . . . . . . 8
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → ((𝑔 ∘ ℎ) ∈ (2o
↑𝑚 𝐴) ↔ (𝑔 ∘ ℎ):𝐴⟶2o)) |
27 | 23, 26 | mpbird 166 |
. . . . . . 7
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (𝑔 ∘ ℎ) ∈ (2o
↑𝑚 𝐴)) |
28 | 7, 10, 27 | rspcdva 2818 |
. . . . . 6
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → DECID
∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) |
29 | | f1ofn 5408 |
. . . . . . . . . . . 12
⊢ (ℎ:𝐴–1-1-onto→𝐵 → ℎ Fn 𝐴) |
30 | 29 | ad3antlr 485 |
. . . . . . . . . . 11
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ℎ Fn 𝐴) |
31 | | f1ocnv 5420 |
. . . . . . . . . . . . . 14
⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵–1-1-onto→𝐴) |
32 | | f1of 5407 |
. . . . . . . . . . . . . 14
⊢ (◡ℎ:𝐵–1-1-onto→𝐴 → ◡ℎ:𝐵⟶𝐴) |
33 | 31, 32 | syl 14 |
. . . . . . . . . . . . 13
⊢ (ℎ:𝐴–1-1-onto→𝐵 → ◡ℎ:𝐵⟶𝐴) |
34 | 33 | ad3antlr 485 |
. . . . . . . . . . . 12
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ◡ℎ:𝐵⟶𝐴) |
35 | | simpr 109 |
. . . . . . . . . . . 12
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) |
36 | 34, 35 | ffvelrnd 5596 |
. . . . . . . . . . 11
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (◡ℎ‘𝑦) ∈ 𝐴) |
37 | | fvco2 5530 |
. . . . . . . . . . 11
⊢ ((ℎ Fn 𝐴 ∧ (◡ℎ‘𝑦) ∈ 𝐴) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦)))) |
38 | 30, 36, 37 | syl2anc 409 |
. . . . . . . . . 10
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = (𝑔‘(ℎ‘(◡ℎ‘𝑦)))) |
39 | | fveqeq2 5470 |
. . . . . . . . . . 11
⊢ (𝑥 = (◡ℎ‘𝑦) → (((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o)) |
40 | | simplr 520 |
. . . . . . . . . . 11
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) |
41 | 39, 40, 36 | rspcdva 2818 |
. . . . . . . . . 10
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → ((𝑔 ∘ ℎ)‘(◡ℎ‘𝑦)) = 1o) |
42 | | f1ocnvfv2 5719 |
. . . . . . . . . . . 12
⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (ℎ‘(◡ℎ‘𝑦)) = 𝑦) |
43 | 42 | fveq2d 5465 |
. . . . . . . . . . 11
⊢ ((ℎ:𝐴–1-1-onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦)) |
44 | 43 | ad4ant24 508 |
. . . . . . . . . 10
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (𝑔‘(ℎ‘(◡ℎ‘𝑦))) = (𝑔‘𝑦)) |
45 | 38, 41, 44 | 3eqtr3rd 2196 |
. . . . . . . . 9
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) ∧ 𝑦 ∈ 𝐵) → (𝑔‘𝑦) = 1o) |
46 | 45 | ralrimiva 2527 |
. . . . . . . 8
⊢
(((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) |
47 | 29 | ad3antlr 485 |
. . . . . . . . . . 11
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → ℎ Fn 𝐴) |
48 | | simpr 109 |
. . . . . . . . . . 11
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) |
49 | | fvco2 5530 |
. . . . . . . . . . 11
⊢ ((ℎ Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥))) |
50 | 47, 48, 49 | syl2anc 409 |
. . . . . . . . . 10
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = (𝑔‘(ℎ‘𝑥))) |
51 | | fveqeq2 5470 |
. . . . . . . . . . 11
⊢ (𝑦 = (ℎ‘𝑥) → ((𝑔‘𝑦) = 1o ↔ (𝑔‘(ℎ‘𝑥)) = 1o)) |
52 | | simplr 520 |
. . . . . . . . . . 11
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) |
53 | 21 | ffvelrnda 5595 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ 𝑥 ∈ 𝐴) → (ℎ‘𝑥) ∈ 𝐵) |
54 | 53 | adantlr 469 |
. . . . . . . . . . 11
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → (ℎ‘𝑥) ∈ 𝐵) |
55 | 51, 52, 54 | rspcdva 2818 |
. . . . . . . . . 10
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → (𝑔‘(ℎ‘𝑥)) = 1o) |
56 | 50, 55 | eqtrd 2187 |
. . . . . . . . 9
⊢
((((((𝐴 ≈
𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) ∧ 𝑥 ∈ 𝐴) → ((𝑔 ∘ ℎ)‘𝑥) = 1o) |
57 | 56 | ralrimiva 2527 |
. . . . . . . 8
⊢
(((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) → ∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o) |
58 | 46, 57 | impbida 586 |
. . . . . . 7
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)) |
59 | 58 | dcbid 824 |
. . . . . 6
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → (DECID
∀𝑥 ∈ 𝐴 ((𝑔 ∘ ℎ)‘𝑥) = 1o ↔ DECID
∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)) |
60 | 28, 59 | mpbid 146 |
. . . . 5
⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) ∧ ℎ:𝐴–1-1-onto→𝐵) → DECID
∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) |
61 | 3, 60 | exlimddv 1875 |
. . . 4
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) ∧ 𝑔 ∈ (2o
↑𝑚 𝐵)) → DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) |
62 | 61 | ralrimiva 2527 |
. . 3
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → ∀𝑔 ∈ (2o
↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o) |
63 | | iswomnimap 7088 |
. . . . 5
⊢ (𝐵 ∈ V → (𝐵 ∈ WOmni ↔
∀𝑔 ∈
(2o ↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)) |
64 | 14, 63 | syl 14 |
. . . 4
⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ WOmni ↔ ∀𝑔 ∈ (2o
↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)) |
65 | 64 | adantr 274 |
. . 3
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → (𝐵 ∈ WOmni ↔ ∀𝑔 ∈ (2o
↑𝑚 𝐵)DECID ∀𝑦 ∈ 𝐵 (𝑔‘𝑦) = 1o)) |
66 | 62, 65 | mpbird 166 |
. 2
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ WOmni) → 𝐵 ∈ WOmni) |
67 | 66 | ex 114 |
1
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni)) |