Theorem isof1oopb 7356

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⟶1-1onto𝐵 ↔ 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 6921	. . . . . . . . 9 ⊢ (𝐻‘𝑥) ∈ V
2		fvex 6921	. . . . . . . . 9 ⊢ (𝐻‘𝑦) ∈ V
3	1, 2	opelvv 5695	. . . . . . . 8 ⊢ ⟨(𝐻‘𝑥), (𝐻‘𝑦)⟩ ∈ (V × V)
4		df-br 5107	. . . . . . . 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 5696	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V))
8		df-br 5107	. . . . . . . 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⟶1-1onto𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)))
13	12	ralrimivva 3209	. . 3 ⊢ (𝐻:𝐴⟶1-1onto𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)))
14	13	pm4.71i 569	. 2 ⊢ (𝐻:𝐴⟶1-1onto𝐵 ↔ (𝐻:𝐴⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))))
15		df-isom 6567	. 2 ⊢ (𝐻 Isom (V × V), (V × V)(𝐴, 𝐵) ↔ (𝐻:𝐴⟶1-1onto𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦))))
16	14, 15	bitr4i 281	1 ⊢ (𝐻:𝐴⟶1-1onto𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ⟨cop 4838   class class class wbr 5106   × cxp 5653  ⟶1-1ontowf1o 6557  ‘cfv 6558   Isom wiso 6559

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-iota 6512  df-fv 6566  df-isom 6567

This theorem is used by: (None)