Theorem isof1oidb 7330

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 6821	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
2		f1fveq 7264	. . . . . 6 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) = (𝐻‘𝑦) ↔ 𝑥 = 𝑦))
3	1, 2	sylan 592	. . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐻‘𝑥) = (𝐻‘𝑦) ↔ 𝑥 = 𝑦))
4		fvex 6896	. . . . . . 7 ⊢ (𝐻‘𝑦) ∈ V
5	4	ideq 5830	. . . . . 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 5829	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
8	7	ad2antll 742	. . . . 5 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 I 𝑦 ↔ 𝑥 = 𝑦))
9	3, 6, 8	3bitr4rd 315	. . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦)))
10	9	ralrimivva 3206	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 I 𝑦 ↔ (𝐻‘𝑥) I (𝐻‘𝑦)))
11	10	pm4.71i 569	. 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 6546	. 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 281	1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103   I cid 5545  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-f1o 6544  df-fv 6545  df-isom 6546

This theorem is used by: (None)