Theorem isof1oopb 7325

Description: A function is a bijection iff it is an isomorphism regarding the universal class of ordered pairs as relations. (Contributed by AV, 9-May-2021.)

Assertion

Ref	Expression
isof1oopb	⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))

Proof of Theorem isof1oopb

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

Step	Hyp	Ref	Expression
1		fvex 6890	. . . . . . . . 9 ⊢ (𝐻‘𝑥) ∈ V
2		fvex 6890	. . . . . . . . 9 ⊢ (𝐻‘𝑦) ∈ V
3	1, 2	opelvv 5691	. . . . . . . 8 ⊢ ⟨(𝐻‘𝑥), (𝐻‘𝑦)⟩ ∈ (V × V)
4		df-br 5104	. . . . . . . 8 ⊢ ((𝐻‘𝑥)(V × V)(𝐻‘𝑦) ↔ ⟨(𝐻‘𝑥), (𝐻‘𝑦)⟩ ∈ (V × V))
5	3, 4	mpbir 234	. . . . . . 7 ⊢ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)
6	5	a1i 11	. . . . . 6 ⊢ (𝑥(V × V)𝑦 → (𝐻‘𝑥)(V × V)(𝐻‘𝑦))
7		opelvvg 5692	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V))
8		df-br 5104	. . . . . . . 8 ⊢ (𝑥(V × V)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V))
9	7, 8	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥(V × V)𝑦)
10	9	a1d 26	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐻‘𝑥)(V × V)(𝐻‘𝑦) → 𝑥(V × V)𝑦))
11	6, 10	impbid2 229	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)))
12	11	adantl 487	. . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)))
13	12	ralrimivva 3206	. . 3 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)))
14	13	pm4.71i 569	. 2 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))))
15		df-isom 6540	. 2 ⊢ (𝐻 Isom (V × V), (V × V)(𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))))
16	14, 15	bitr4i 281	1 ⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   × cxp 5649  –1-1-onto→wf1o 6530  ‘cfv 6531   Isom wiso 6532

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-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-sb 2100  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-xp 5657  df-iota 6487  df-fv 6539  df-isom 6540

This theorem is used by: (None)