Step | Hyp | Ref
| Expression |
1 | | simprl 790 |
. . . . 5
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝐻:𝐴–onto→𝐵) |
2 | | fof 6028 |
. . . . 5
⊢ (𝐻:𝐴–onto→𝐵 → 𝐻:𝐴⟶𝐵) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝐻:𝐴⟶𝐵) |
4 | | simpll 786 |
. . . . . . . 8
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝑅 Or 𝐴) |
5 | | sotrieq 4986 |
. . . . . . . . 9
⊢ ((𝑅 Or 𝐴 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎 = 𝑏 ↔ ¬ (𝑎𝑅𝑏 ∨ 𝑏𝑅𝑎))) |
6 | 5 | con2bid 343 |
. . . . . . . 8
⊢ ((𝑅 Or 𝐴 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝑎𝑅𝑏 ∨ 𝑏𝑅𝑎) ↔ ¬ 𝑎 = 𝑏)) |
7 | 4, 6 | sylan 487 |
. . . . . . 7
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝑎𝑅𝑏 ∨ 𝑏𝑅𝑎) ↔ ¬ 𝑎 = 𝑏)) |
8 | | simprr 792 |
. . . . . . . . . 10
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))) |
9 | | breq1 4586 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑎 → (𝑥𝑅𝑦 ↔ 𝑎𝑅𝑦)) |
10 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑎 → (𝐻‘𝑥) = (𝐻‘𝑎)) |
11 | 10 | breq1d 4593 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑎 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑦))) |
12 | 9, 11 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑎 → ((𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑎𝑅𝑦 → (𝐻‘𝑎)𝑆(𝐻‘𝑦)))) |
13 | | breq2 4587 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑏 → (𝑎𝑅𝑦 ↔ 𝑎𝑅𝑏)) |
14 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑏 → (𝐻‘𝑦) = (𝐻‘𝑏)) |
15 | 14 | breq2d 4595 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑏 → ((𝐻‘𝑎)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
16 | 13, 15 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑏 → ((𝑎𝑅𝑦 → (𝐻‘𝑎)𝑆(𝐻‘𝑦)) ↔ (𝑎𝑅𝑏 → (𝐻‘𝑎)𝑆(𝐻‘𝑏)))) |
17 | 12, 16 | rspc2va 3294 |
. . . . . . . . . . 11
⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → (𝑎𝑅𝑏 → (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
18 | 17 | ancoms 468 |
. . . . . . . . . 10
⊢
((∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 → (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
19 | 8, 18 | sylan 487 |
. . . . . . . . 9
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 → (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
20 | | simpllr 795 |
. . . . . . . . . . 11
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑆 Po 𝐵) |
21 | | simplrl 796 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝐻:𝐴–onto→𝐵) |
22 | 21, 2 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝐻:𝐴⟶𝐵) |
23 | | simprr 792 |
. . . . . . . . . . . 12
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏 ∈ 𝐴) |
24 | 22, 23 | ffvelrnd 6268 |
. . . . . . . . . . 11
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝐻‘𝑏) ∈ 𝐵) |
25 | | poirr 4970 |
. . . . . . . . . . . 12
⊢ ((𝑆 Po 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵) → ¬ (𝐻‘𝑏)𝑆(𝐻‘𝑏)) |
26 | | breq1 4586 |
. . . . . . . . . . . . 13
⊢ ((𝐻‘𝑎) = (𝐻‘𝑏) → ((𝐻‘𝑎)𝑆(𝐻‘𝑏) ↔ (𝐻‘𝑏)𝑆(𝐻‘𝑏))) |
27 | 26 | notbid 307 |
. . . . . . . . . . . 12
⊢ ((𝐻‘𝑎) = (𝐻‘𝑏) → (¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏) ↔ ¬ (𝐻‘𝑏)𝑆(𝐻‘𝑏))) |
28 | 25, 27 | syl5ibrcom 236 |
. . . . . . . . . . 11
⊢ ((𝑆 Po 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵) → ((𝐻‘𝑎) = (𝐻‘𝑏) → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
29 | 20, 24, 28 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐻‘𝑎) = (𝐻‘𝑏) → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
30 | 29 | con2d 128 |
. . . . . . . . 9
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐻‘𝑎)𝑆(𝐻‘𝑏) → ¬ (𝐻‘𝑎) = (𝐻‘𝑏))) |
31 | 19, 30 | syld 46 |
. . . . . . . 8
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 → ¬ (𝐻‘𝑎) = (𝐻‘𝑏))) |
32 | | breq1 4586 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑏 → (𝑥𝑅𝑦 ↔ 𝑏𝑅𝑦)) |
33 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑏 → (𝐻‘𝑥) = (𝐻‘𝑏)) |
34 | 33 | breq1d 4593 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑏 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑏)𝑆(𝐻‘𝑦))) |
35 | 32, 34 | imbi12d 333 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑏 → ((𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝑏𝑅𝑦 → (𝐻‘𝑏)𝑆(𝐻‘𝑦)))) |
36 | | breq2 4587 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑎 → (𝑏𝑅𝑦 ↔ 𝑏𝑅𝑎)) |
37 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑎 → (𝐻‘𝑦) = (𝐻‘𝑎)) |
38 | 37 | breq2d 4595 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑎 → ((𝐻‘𝑏)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝑏)𝑆(𝐻‘𝑎))) |
39 | 36, 38 | imbi12d 333 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑎 → ((𝑏𝑅𝑦 → (𝐻‘𝑏)𝑆(𝐻‘𝑦)) ↔ (𝑏𝑅𝑎 → (𝐻‘𝑏)𝑆(𝐻‘𝑎)))) |
40 | 35, 39 | rspc2va 3294 |
. . . . . . . . . . . 12
⊢ (((𝑏 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → (𝑏𝑅𝑎 → (𝐻‘𝑏)𝑆(𝐻‘𝑎))) |
41 | 40 | ancoms 468 |
. . . . . . . . . . 11
⊢
((∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ∧ (𝑏 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴)) → (𝑏𝑅𝑎 → (𝐻‘𝑏)𝑆(𝐻‘𝑎))) |
42 | 41 | ancom2s 840 |
. . . . . . . . . 10
⊢
((∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑏𝑅𝑎 → (𝐻‘𝑏)𝑆(𝐻‘𝑎))) |
43 | 8, 42 | sylan 487 |
. . . . . . . . 9
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑏𝑅𝑎 → (𝐻‘𝑏)𝑆(𝐻‘𝑎))) |
44 | | breq2 4587 |
. . . . . . . . . . . . 13
⊢ ((𝐻‘𝑎) = (𝐻‘𝑏) → ((𝐻‘𝑏)𝑆(𝐻‘𝑎) ↔ (𝐻‘𝑏)𝑆(𝐻‘𝑏))) |
45 | 44 | notbid 307 |
. . . . . . . . . . . 12
⊢ ((𝐻‘𝑎) = (𝐻‘𝑏) → (¬ (𝐻‘𝑏)𝑆(𝐻‘𝑎) ↔ ¬ (𝐻‘𝑏)𝑆(𝐻‘𝑏))) |
46 | 25, 45 | syl5ibrcom 236 |
. . . . . . . . . . 11
⊢ ((𝑆 Po 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵) → ((𝐻‘𝑎) = (𝐻‘𝑏) → ¬ (𝐻‘𝑏)𝑆(𝐻‘𝑎))) |
47 | 20, 24, 46 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐻‘𝑎) = (𝐻‘𝑏) → ¬ (𝐻‘𝑏)𝑆(𝐻‘𝑎))) |
48 | 47 | con2d 128 |
. . . . . . . . 9
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐻‘𝑏)𝑆(𝐻‘𝑎) → ¬ (𝐻‘𝑎) = (𝐻‘𝑏))) |
49 | 43, 48 | syld 46 |
. . . . . . . 8
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑏𝑅𝑎 → ¬ (𝐻‘𝑎) = (𝐻‘𝑏))) |
50 | 31, 49 | jaod 394 |
. . . . . . 7
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝑎𝑅𝑏 ∨ 𝑏𝑅𝑎) → ¬ (𝐻‘𝑎) = (𝐻‘𝑏))) |
51 | 7, 50 | sylbird 249 |
. . . . . 6
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (¬ 𝑎 = 𝑏 → ¬ (𝐻‘𝑎) = (𝐻‘𝑏))) |
52 | 51 | con4d 113 |
. . . . 5
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐻‘𝑎) = (𝐻‘𝑏) → 𝑎 = 𝑏)) |
53 | 52 | ralrimivva 2954 |
. . . 4
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ((𝐻‘𝑎) = (𝐻‘𝑏) → 𝑎 = 𝑏)) |
54 | | dff13 6416 |
. . . 4
⊢ (𝐻:𝐴–1-1→𝐵 ↔ (𝐻:𝐴⟶𝐵 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ((𝐻‘𝑎) = (𝐻‘𝑏) → 𝑎 = 𝑏))) |
55 | 3, 53, 54 | sylanbrc 695 |
. . 3
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝐻:𝐴–1-1→𝐵) |
56 | | df-f1o 5811 |
. . 3
⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ (𝐻:𝐴–1-1→𝐵 ∧ 𝐻:𝐴–onto→𝐵)) |
57 | 55, 1, 56 | sylanbrc 695 |
. 2
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝐻:𝐴–1-1-onto→𝐵) |
58 | | sotric 4985 |
. . . . . . 7
⊢ ((𝑅 Or 𝐴 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 ↔ ¬ (𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))) |
59 | 58 | con2bid 343 |
. . . . . 6
⊢ ((𝑅 Or 𝐴 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) ↔ ¬ 𝑎𝑅𝑏)) |
60 | 4, 59 | sylan 487 |
. . . . 5
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) ↔ ¬ 𝑎𝑅𝑏)) |
61 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (𝐻‘𝑎) = (𝐻‘𝑏)) |
62 | 61 | breq1d 4593 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → ((𝐻‘𝑎)𝑆(𝐻‘𝑏) ↔ (𝐻‘𝑏)𝑆(𝐻‘𝑏))) |
63 | 62 | notbid 307 |
. . . . . . . 8
⊢ (𝑎 = 𝑏 → (¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏) ↔ ¬ (𝐻‘𝑏)𝑆(𝐻‘𝑏))) |
64 | 25, 63 | syl5ibrcom 236 |
. . . . . . 7
⊢ ((𝑆 Po 𝐵 ∧ (𝐻‘𝑏) ∈ 𝐵) → (𝑎 = 𝑏 → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
65 | 20, 24, 64 | syl2anc 691 |
. . . . . 6
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎 = 𝑏 → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
66 | | simprl 790 |
. . . . . . . . 9
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑎 ∈ 𝐴) |
67 | 22, 66 | ffvelrnd 6268 |
. . . . . . . 8
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝐻‘𝑎) ∈ 𝐵) |
68 | | po2nr 4972 |
. . . . . . . . 9
⊢ ((𝑆 Po 𝐵 ∧ ((𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑎) ∈ 𝐵)) → ¬ ((𝐻‘𝑏)𝑆(𝐻‘𝑎) ∧ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
69 | | imnan 437 |
. . . . . . . . 9
⊢ (((𝐻‘𝑏)𝑆(𝐻‘𝑎) → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) ↔ ¬ ((𝐻‘𝑏)𝑆(𝐻‘𝑎) ∧ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
70 | 68, 69 | sylibr 223 |
. . . . . . . 8
⊢ ((𝑆 Po 𝐵 ∧ ((𝐻‘𝑏) ∈ 𝐵 ∧ (𝐻‘𝑎) ∈ 𝐵)) → ((𝐻‘𝑏)𝑆(𝐻‘𝑎) → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
71 | 20, 24, 67, 70 | syl12anc 1316 |
. . . . . . 7
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝐻‘𝑏)𝑆(𝐻‘𝑎) → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
72 | 43, 71 | syld 46 |
. . . . . 6
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑏𝑅𝑎 → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
73 | 65, 72 | jaod 394 |
. . . . 5
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
74 | 60, 73 | sylbird 249 |
. . . 4
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (¬ 𝑎𝑅𝑏 → ¬ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
75 | 19, 74 | impcon4bid 216 |
. . 3
⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
76 | 75 | ralrimivva 2954 |
. 2
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) |
77 | | df-isom 5813 |
. 2
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))) |
78 | 57, 76, 77 | sylanbrc 695 |
1
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) |