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 6896	. . . . . . . . 9 ⊢ (𝐻‘𝑥) ∈ V
2		fvex 6896	. . . . . . . . 9 ⊢ (𝐻‘𝑦) ∈ V
3	1, 2	opelvv 5703	. . . . . . . 8 ⊢ ⟨(𝐻‘𝑥), (𝐻‘𝑦)⟩ ∈ (V × V)
4		df-br 5111	. . . . . . . 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 5704	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V))
8		df-br 5111	. . . . . . . 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 486	. . . 4 ⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻‘𝑥)(V × V)(𝐻‘𝑦)))
13	12	ralrimivva 3208	. . 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 568	. 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 6547	. 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))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  ⟨cop 4596   class class class wbr 5110   × cxp 5661  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-iota 6494  df-fv 6546  df-isom 6547

This theorem is referenced by: (None)