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Mirrors > Home > MPE Home > Th. List > isof1oidb | Structured version Visualization version GIF version |
Description: A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021.) |
Ref | Expression |
---|---|
isof1oidb | ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1of1 6616 | . . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵) | |
2 | f1fveq 7022 | . . . . . 6 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) = (𝐻‘𝑦) ↔ 𝑥 = 𝑦)) | |
3 | 1, 2 | sylan 582 | . . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) = (𝐻‘𝑦) ↔ 𝑥 = 𝑦)) |
4 | fvex 6685 | . . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V | |
5 | 4 | ideq 5725 | . . . . . 6 ⊢ ((𝐻‘𝑥) I (𝐻‘𝑦) ↔ (𝐻‘𝑥) = (𝐻‘𝑦)) |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) I (𝐻‘𝑦) ↔ (𝐻‘𝑥) = (𝐻‘𝑦))) |
7 | ideqg 5724 | . . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) | |
8 | 7 | ad2antll 727 | . . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)) |
9 | 3, 6, 8 | 3bitr4rd 314 | . . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦))) |
10 | 9 | ralrimivva 3193 | . . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦))) |
11 | 10 | pm4.71i 562 | . 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦)))) |
12 | df-isom 6366 | . 2 ⊢ (𝐻 Isom I , I (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦)))) | |
13 | 11, 12 | bitr4i 280 | 1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 class class class wbr 5068 I cid 5461 –1-1→wf1 6354 –1-1-onto→wf1o 6356 ‘cfv 6357 Isom wiso 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-f1o 6364 df-fv 6365 df-isom 6366 |
This theorem is referenced by: (None) |
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