| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | f1ocnv 6859 | . . . 4
⊢ (𝐻:𝐴–1-1-onto→𝐵 → ◡𝐻:𝐵–1-1-onto→𝐴) | 
| 2 | 1 | adantr 480 | . . 3
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ◡𝐻:𝐵–1-1-onto→𝐴) | 
| 3 |  | f1ocnvfv2 7298 | . . . . . . . 8
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑧 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑧)) = 𝑧) | 
| 4 | 3 | adantrr 717 | . . . . . . 7
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐻‘(◡𝐻‘𝑧)) = 𝑧) | 
| 5 |  | f1ocnvfv2 7298 | . . . . . . . 8
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑤 ∈ 𝐵) → (𝐻‘(◡𝐻‘𝑤)) = 𝑤) | 
| 6 | 5 | adantrl 716 | . . . . . . 7
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝐻‘(◡𝐻‘𝑤)) = 𝑤) | 
| 7 | 4, 6 | breq12d 5155 | . . . . . 6
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ 𝑧𝑆𝑤)) | 
| 8 | 7 | adantlr 715 | . . . . 5
⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ 𝑧𝑆𝑤)) | 
| 9 |  | f1of 6847 | . . . . . . 7
⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → ◡𝐻:𝐵⟶𝐴) | 
| 10 | 1, 9 | syl 17 | . . . . . 6
⊢ (𝐻:𝐴–1-1-onto→𝐵 → ◡𝐻:𝐵⟶𝐴) | 
| 11 |  | ffvelcdm 7100 | . . . . . . . . 9
⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → (◡𝐻‘𝑧) ∈ 𝐴) | 
| 12 |  | ffvelcdm 7100 | . . . . . . . . 9
⊢ ((◡𝐻:𝐵⟶𝐴 ∧ 𝑤 ∈ 𝐵) → (◡𝐻‘𝑤) ∈ 𝐴) | 
| 13 | 11, 12 | anim12dan 619 | . . . . . . . 8
⊢ ((◡𝐻:𝐵⟶𝐴 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((◡𝐻‘𝑧) ∈ 𝐴 ∧ (◡𝐻‘𝑤) ∈ 𝐴)) | 
| 14 |  | breq1 5145 | . . . . . . . . . . 11
⊢ (𝑥 = (◡𝐻‘𝑧) → (𝑥𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅𝑦)) | 
| 15 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑥 = (◡𝐻‘𝑧) → (𝐻‘𝑥) = (𝐻‘(◡𝐻‘𝑧))) | 
| 16 | 15 | breq1d 5152 | . . . . . . . . . . 11
⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦))) | 
| 17 | 14, 16 | bibi12d 345 | . . . . . . . . . 10
⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ((◡𝐻‘𝑧)𝑅𝑦 ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦)))) | 
| 18 |  | bicom 222 | . . . . . . . . . 10
⊢ (((◡𝐻‘𝑧)𝑅𝑦 ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦)) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦)) | 
| 19 | 17, 18 | bitrdi 287 | . . . . . . . . 9
⊢ (𝑥 = (◡𝐻‘𝑧) → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦))) | 
| 20 |  | fveq2 6905 | . . . . . . . . . . 11
⊢ (𝑦 = (◡𝐻‘𝑤) → (𝐻‘𝑦) = (𝐻‘(◡𝐻‘𝑤))) | 
| 21 | 20 | breq2d 5154 | . . . . . . . . . 10
⊢ (𝑦 = (◡𝐻‘𝑤) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)))) | 
| 22 |  | breq2 5146 | . . . . . . . . . 10
⊢ (𝑦 = (◡𝐻‘𝑤) → ((◡𝐻‘𝑧)𝑅𝑦 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))) | 
| 23 | 21, 22 | bibi12d 345 | . . . . . . . . 9
⊢ (𝑦 = (◡𝐻‘𝑤) → (((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘𝑦) ↔ (◡𝐻‘𝑧)𝑅𝑦) ↔ ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))) | 
| 24 | 19, 23 | rspc2va 3633 | . . . . . . . 8
⊢ ((((◡𝐻‘𝑧) ∈ 𝐴 ∧ (◡𝐻‘𝑤) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))) | 
| 25 | 13, 24 | sylan 580 | . . . . . . 7
⊢ (((◡𝐻:𝐵⟶𝐴 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))) | 
| 26 | 25 | an32s 652 | . . . . . 6
⊢ (((◡𝐻:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))) | 
| 27 | 10, 26 | sylanl1 680 | . . . . 5
⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝐻‘(◡𝐻‘𝑧))𝑆(𝐻‘(◡𝐻‘𝑤)) ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))) | 
| 28 | 8, 27 | bitr3d 281 | . . . 4
⊢ (((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧𝑆𝑤 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))) | 
| 29 | 28 | ralrimivva 3201 | . . 3
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤))) | 
| 30 | 2, 29 | jca 511 | . 2
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) → (◡𝐻:𝐵–1-1-onto→𝐴 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))) | 
| 31 |  | df-isom 6569 | . 2
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | 
| 32 |  | df-isom 6569 | . 2
⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) ↔ (◡𝐻:𝐵–1-1-onto→𝐴 ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝑆𝑤 ↔ (◡𝐻‘𝑧)𝑅(◡𝐻‘𝑤)))) | 
| 33 | 30, 31, 32 | 3imtr4i 292 | 1
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴)) |