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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:𝐴1-1-onto𝐵𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))

Proof of Theorem isof1oopb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6905 . . . . . . . . 9 (𝐻𝑥) ∈ V
2 fvex 6905 . . . . . . . . 9 (𝐻𝑦) ∈ V
31, 2opelvv 5717 . . . . . . . 8 ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V)
4 df-br 5150 . . . . . . . 8 ((𝐻𝑥)(V × V)(𝐻𝑦) ↔ ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V))
53, 4mpbir 230 . . . . . . 7 (𝐻𝑥)(V × V)(𝐻𝑦)
65a1i 11 . . . . . 6 (𝑥(V × V)𝑦 → (𝐻𝑥)(V × V)(𝐻𝑦))
7 opelvvg 5718 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V))
8 df-br 5150 . . . . . . . 8 (𝑥(V × V)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V))
97, 8sylibr 233 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → 𝑥(V × V)𝑦)
109a1d 25 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ((𝐻𝑥)(V × V)(𝐻𝑦) → 𝑥(V × V)𝑦))
116, 10impbid2 225 . . . . 5 ((𝑥𝐴𝑦𝐴) → (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1211adantl 481 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1312ralrimivva 3199 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1413pm4.71i 559 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
15 df-isom 6553 . 2 (𝐻 Isom (V × V), (V × V)(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
1614, 15bitr4i 277 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2105  wral 3060  Vcvv 3473  cop 4635   class class class wbr 5149   × cxp 5675  1-1-ontowf1o 6543  cfv 6544   Isom wiso 6545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-iota 6496  df-fv 6552  df-isom 6553
This theorem is referenced by: (None)
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