Step | Hyp | Ref
| Expression |
1 | | ordsson 6989 |
. . . . . 6
⊢ (Ord
𝐴 → 𝐴 ⊆ On) |
2 | 1 | 3ad2ant2 1083 |
. . . . 5
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 ⊆ On) |
3 | 2 | sseld 3602 |
. . . 4
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ On)) |
4 | | eleq1 2689 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
5 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) |
6 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) |
7 | 5, 6 | eqeq12d 2637 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑦) = 𝑦)) |
8 | 4, 7 | imbi12d 334 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥) ↔ (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦))) |
9 | 8 | imbi2d 330 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥)) ↔ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)))) |
10 | | r19.21v 2960 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝑥 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)) ↔ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦))) |
11 | | ordelss 5739 |
. . . . . . . . . . . . . . . 16
⊢ ((Ord
𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴) |
12 | 11 | 3ad2antl2 1224 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴) |
13 | 12 | sselda 3603 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴) |
14 | | pm5.5 351 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) ↔ (𝐹‘𝑦) = 𝑦)) |
15 | 13, 14 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) ↔ (𝐹‘𝑦) = 𝑦)) |
16 | 15 | ralbidva 2985 |
. . . . . . . . . . . 12
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) ↔ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) |
17 | | isof1o 6573 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵) |
18 | 17 | 3ad2ant1 1082 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐹:𝐴–1-1-onto→𝐵) |
19 | 18 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝐹:𝐴–1-1-onto→𝐵) |
20 | | simpll3 1102 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → Ord 𝐵) |
21 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑧 ∈ (𝐹‘𝑥)) |
22 | | f1of 6137 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) |
23 | 17, 22 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴⟶𝐵) |
24 | 23 | 3ad2ant1 1082 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐹:𝐴⟶𝐵) |
25 | 24 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝐹:𝐴⟶𝐵) |
26 | | simplrl 800 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑥 ∈ 𝐴) |
27 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
28 | 25, 26, 27 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝐹‘𝑥) ∈ 𝐵) |
29 | 21, 28 | jca 554 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝑧 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝐵)) |
30 | | ordtr1 5767 |
. . . . . . . . . . . . . . . . . . 19
⊢ (Ord
𝐵 → ((𝑧 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ 𝐵) → 𝑧 ∈ 𝐵)) |
31 | 20, 29, 30 | sylc 65 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑧 ∈ 𝐵) |
32 | | f1ocnvfv2 6533 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧) |
33 | 19, 31, 32 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧) |
34 | 33, 21 | eqeltrd 2701 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹‘𝑥)) |
35 | | simpll1 1100 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝐹 Isom E , E (𝐴, 𝐵)) |
36 | | f1ocnv 6149 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) |
37 | | f1of 6137 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) |
38 | 19, 36, 37 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → ◡𝐹:𝐵⟶𝐴) |
39 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((◡𝐹:𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → (◡𝐹‘𝑧) ∈ 𝐴) |
40 | 38, 31, 39 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (◡𝐹‘𝑧) ∈ 𝐴) |
41 | | isorel 6576 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ ((◡𝐹‘𝑧) ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((◡𝐹‘𝑧) E 𝑥 ↔ (𝐹‘(◡𝐹‘𝑧)) E (𝐹‘𝑥))) |
42 | 35, 40, 26, 41 | syl12anc 1324 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → ((◡𝐹‘𝑧) E 𝑥 ↔ (𝐹‘(◡𝐹‘𝑧)) E (𝐹‘𝑥))) |
43 | | vex 3203 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑥 ∈ V |
44 | 43 | epelc 5031 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((◡𝐹‘𝑧) E 𝑥 ↔ (◡𝐹‘𝑧) ∈ 𝑥) |
45 | | fvex 6201 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐹‘𝑥) ∈ V |
46 | 45 | epelc 5031 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹‘(◡𝐹‘𝑧)) E (𝐹‘𝑥) ↔ (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹‘𝑥)) |
47 | 42, 44, 46 | 3bitr3g 302 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → ((◡𝐹‘𝑧) ∈ 𝑥 ↔ (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹‘𝑥))) |
48 | 34, 47 | mpbird 247 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (◡𝐹‘𝑧) ∈ 𝑥) |
49 | | simplrr 801 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦) |
50 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (◡𝐹‘𝑧) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑧))) |
51 | | id 22 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (◡𝐹‘𝑧) → 𝑦 = (◡𝐹‘𝑧)) |
52 | 50, 51 | eqeq12d 2637 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (◡𝐹‘𝑧) → ((𝐹‘𝑦) = 𝑦 ↔ (𝐹‘(◡𝐹‘𝑧)) = (◡𝐹‘𝑧))) |
53 | 52 | rspcv 3305 |
. . . . . . . . . . . . . . . . . 18
⊢ ((◡𝐹‘𝑧) ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦 → (𝐹‘(◡𝐹‘𝑧)) = (◡𝐹‘𝑧))) |
54 | 48, 49, 53 | sylc 65 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → (𝐹‘(◡𝐹‘𝑧)) = (◡𝐹‘𝑧)) |
55 | 33, 54 | eqtr3d 2658 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑧 = (◡𝐹‘𝑧)) |
56 | 55, 48 | eqeltrd 2701 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ (𝐹‘𝑥)) → 𝑧 ∈ 𝑥) |
57 | | simprr 796 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦) |
58 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) |
59 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧) |
60 | 58, 59 | eqeq12d 2637 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) = 𝑦 ↔ (𝐹‘𝑧) = 𝑧)) |
61 | 60 | rspccva 3308 |
. . . . . . . . . . . . . . . . 17
⊢
((∀𝑦 ∈
𝑥 (𝐹‘𝑦) = 𝑦 ∧ 𝑧 ∈ 𝑥) → (𝐹‘𝑧) = 𝑧) |
62 | 57, 61 | sylan 488 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → (𝐹‘𝑧) = 𝑧) |
63 | | epel 5032 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 E 𝑥 ↔ 𝑧 ∈ 𝑥) |
64 | 63 | biimpri 218 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ 𝑥 → 𝑧 E 𝑥) |
65 | 64 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝑧 E 𝑥) |
66 | | simpll1 1100 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝐹 Isom E , E (𝐴, 𝐵)) |
67 | | simpl2 1065 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → Ord 𝐴) |
68 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → 𝑥 ∈ 𝐴) |
69 | 67, 68, 11 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → 𝑥 ⊆ 𝐴) |
70 | 69 | sselda 3603 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝐴) |
71 | | simplrl 800 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ 𝐴) |
72 | | isorel 6576 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑧 E 𝑥 ↔ (𝐹‘𝑧) E (𝐹‘𝑥))) |
73 | 66, 70, 71, 72 | syl12anc 1324 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → (𝑧 E 𝑥 ↔ (𝐹‘𝑧) E (𝐹‘𝑥))) |
74 | 65, 73 | mpbid 222 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → (𝐹‘𝑧) E (𝐹‘𝑥)) |
75 | 45 | epelc 5031 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹‘𝑧) E (𝐹‘𝑥) ↔ (𝐹‘𝑧) ∈ (𝐹‘𝑥)) |
76 | 74, 75 | sylib 208 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → (𝐹‘𝑧) ∈ (𝐹‘𝑥)) |
77 | 62, 76 | eqeltrrd 2702 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ (𝐹‘𝑥)) |
78 | 56, 77 | impbida 877 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → (𝑧 ∈ (𝐹‘𝑥) ↔ 𝑧 ∈ 𝑥)) |
79 | 78 | eqrdv 2620 |
. . . . . . . . . . . . 13
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦)) → (𝐹‘𝑥) = 𝑥) |
80 | 79 | expr 643 |
. . . . . . . . . . . 12
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) = 𝑦 → (𝐹‘𝑥) = 𝑥)) |
81 | 16, 80 | sylbid 230 |
. . . . . . . . . . 11
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) → (𝐹‘𝑥) = 𝑥)) |
82 | 81 | ex 450 |
. . . . . . . . . 10
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) → (𝐹‘𝑥) = 𝑥))) |
83 | 82 | com23 86 |
. . . . . . . . 9
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥))) |
84 | 83 | a2i 14 |
. . . . . . . 8
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)) → ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥))) |
85 | 84 | a1i 11 |
. . . . . . 7
⊢ (𝑥 ∈ On → (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)) → ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥)))) |
86 | 10, 85 | syl5bi 232 |
. . . . . 6
⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑦 ∈ 𝐴 → (𝐹‘𝑦) = 𝑦)) → ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥)))) |
87 | 9, 86 | tfis2 7056 |
. . . . 5
⊢ (𝑥 ∈ On → ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥))) |
88 | 87 | com3l 89 |
. . . 4
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ On → (𝐹‘𝑥) = 𝑥))) |
89 | 3, 88 | mpdd 43 |
. . 3
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = 𝑥)) |
90 | 89 | ralrimiv 2965 |
. 2
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) |
91 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) |
92 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧) |
93 | 91, 92 | eqeq12d 2637 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑧) = 𝑧)) |
94 | 93 | rspccva 3308 |
. . . . . . 7
⊢
((∀𝑥 ∈
𝐴 (𝐹‘𝑥) = 𝑥 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝑧) |
95 | 94 | adantll 750 |
. . . . . 6
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝑧) |
96 | | ffvelrn 6357 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
97 | 23, 96 | sylan 488 |
. . . . . . . 8
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
98 | 97 | 3ad2antl1 1223 |
. . . . . . 7
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
99 | 98 | adantlr 751 |
. . . . . 6
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
100 | 95, 99 | eqeltrrd 2702 |
. . . . 5
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐵) |
101 | 100 | ex 450 |
. . . 4
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)) |
102 | | simpl1 1064 |
. . . . . . . 8
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → 𝐹 Isom E , E (𝐴, 𝐵)) |
103 | | f1ofo 6144 |
. . . . . . . . 9
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) |
104 | | forn 6118 |
. . . . . . . . 9
⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) |
105 | 17, 103, 104 | 3syl 18 |
. . . . . . . 8
⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ran 𝐹 = 𝐵) |
106 | 102, 105 | syl 17 |
. . . . . . 7
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → ran 𝐹 = 𝐵) |
107 | 106 | eleq2d 2687 |
. . . . . 6
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ ran 𝐹 ↔ 𝑧 ∈ 𝐵)) |
108 | | f1ofn 6138 |
. . . . . . . . . 10
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴) |
109 | 17, 108 | syl 17 |
. . . . . . . . 9
⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹 Fn 𝐴) |
110 | 109 | 3ad2ant1 1082 |
. . . . . . . 8
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐹 Fn 𝐴) |
111 | 110 | adantr 481 |
. . . . . . 7
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → 𝐹 Fn 𝐴) |
112 | | fvelrnb 6243 |
. . . . . . 7
⊢ (𝐹 Fn 𝐴 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝐴 (𝐹‘𝑤) = 𝑧)) |
113 | 111, 112 | syl 17 |
. . . . . 6
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝐴 (𝐹‘𝑤) = 𝑧)) |
114 | 107, 113 | bitr3d 270 |
. . . . 5
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ 𝐵 ↔ ∃𝑤 ∈ 𝐴 (𝐹‘𝑤) = 𝑧)) |
115 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) |
116 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤) |
117 | 115, 116 | eqeq12d 2637 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑤) = 𝑤)) |
118 | 117 | rspcv 3305 |
. . . . . . . . . 10
⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 → (𝐹‘𝑤) = 𝑤)) |
119 | 118 | a1i 11 |
. . . . . . . . 9
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 → (𝐹‘𝑤) = 𝑤))) |
120 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧) → (𝐹‘𝑤) = 𝑧) |
121 | | simpl 473 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧) → (𝐹‘𝑤) = 𝑤) |
122 | 120, 121 | eqtr3d 2658 |
. . . . . . . . . . . 12
⊢ (((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧) → 𝑧 = 𝑤) |
123 | 122 | adantl 482 |
. . . . . . . . . . 11
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑤 ∈ 𝐴) ∧ ((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧)) → 𝑧 = 𝑤) |
124 | | simplr 792 |
. . . . . . . . . . 11
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑤 ∈ 𝐴) ∧ ((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧)) → 𝑤 ∈ 𝐴) |
125 | 123, 124 | eqeltrd 2701 |
. . . . . . . . . 10
⊢ ((((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ 𝑤 ∈ 𝐴) ∧ ((𝐹‘𝑤) = 𝑤 ∧ (𝐹‘𝑤) = 𝑧)) → 𝑧 ∈ 𝐴) |
126 | 125 | exp43 640 |
. . . . . . . . 9
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑤 ∈ 𝐴 → ((𝐹‘𝑤) = 𝑤 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴)))) |
127 | 119, 126 | syldd 72 |
. . . . . . . 8
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴)))) |
128 | 127 | com23 86 |
. . . . . . 7
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥 → (𝑤 ∈ 𝐴 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴)))) |
129 | 128 | imp 445 |
. . . . . 6
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑤 ∈ 𝐴 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴))) |
130 | 129 | rexlimdv 3030 |
. . . . 5
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (∃𝑤 ∈ 𝐴 (𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐴)) |
131 | 114, 130 | sylbid 230 |
. . . 4
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐴)) |
132 | 101, 131 | impbid 202 |
. . 3
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → (𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) |
133 | 132 | eqrdv 2620 |
. 2
⊢ (((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑥) → 𝐴 = 𝐵) |
134 | 90, 133 | mpdan 702 |
1
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵) |