Theorem isof1oidb 7338

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 6843	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
2		f1fveq 7278	. . . . . 6 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) = (𝐻‘𝑦) ↔ 𝑥 = 𝑦))
3	1, 2	sylan 578	. . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) = (𝐻‘𝑦) ↔ 𝑥 = 𝑦))
4		fvex 6915	. . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V
5	4	ideq 5859	. . . . . 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 5858	. . . . . 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 311	. . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦)))
10	9	ralrimivva 3198	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦)))
11	10	pm4.71i 558	. 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 6562	. 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 277	1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058   class class class wbr 5152   I cid 5579  –1-1→wf1 6550  –1-1-onto→wf1o 6552  ‘cfv 6553   Isom wiso 6554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-f1o 6560  df-fv 6561  df-isom 6562

This theorem is referenced by: (None)