Theorem isof1oidb 7325

Description: A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021.)

Assertion

Ref	Expression
isof1oidb	⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵))

Proof of Theorem isof1oidb

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1of1 6826	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
2		f1fveq 7263	. . . . . 6 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) = (𝐻‘𝑦) ↔ 𝑥 = 𝑦))
3	1, 2	sylan 580	. . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) = (𝐻‘𝑦) ↔ 𝑥 = 𝑦))
4		fvex 6898	. . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V
5	4	ideq 5843	. . . . . 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 5842	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
8	7	ad2antll 729	. . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
9	3, 6, 8	3bitr4rd 312	. . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦)))
10	9	ralrimivva 3189	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦)))
11	10	pm4.71i 559	. 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 6549	. 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 278	1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050   class class class wbr 5123   I cid 5557  –1-1→wf1 6537  –1-1-onto→wf1o 6539  ‘cfv 6540   Isom wiso 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-f1o 6547  df-fv 6548  df-isom 6549

This theorem is referenced by: (None)