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Theorem isof1oopb 7072
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 6682 . . . . . . . . 9 (𝐻𝑥) ∈ V
2 fvex 6682 . . . . . . . . 9 (𝐻𝑦) ∈ V
31, 2opelvv 5593 . . . . . . . 8 ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V)
4 df-br 5064 . . . . . . . 8 ((𝐻𝑥)(V × V)(𝐻𝑦) ↔ ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V))
53, 4mpbir 232 . . . . . . 7 (𝐻𝑥)(V × V)(𝐻𝑦)
65a1i 11 . . . . . 6 (𝑥(V × V)𝑦 → (𝐻𝑥)(V × V)(𝐻𝑦))
7 opelvvg 5594 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V))
8 df-br 5064 . . . . . . . 8 (𝑥(V × V)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V))
97, 8sylibr 235 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → 𝑥(V × V)𝑦)
109a1d 25 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ((𝐻𝑥)(V × V)(𝐻𝑦) → 𝑥(V × V)𝑦))
116, 10impbid2 227 . . . . 5 ((𝑥𝐴𝑦𝐴) → (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1211adantl 482 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1312ralrimivva 3196 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1413pm4.71i 560 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
15 df-isom 6363 . 2 (𝐻 Isom (V × V), (V × V)(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
1614, 15bitr4i 279 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2107  wral 3143  Vcvv 3500  cop 4570   class class class wbr 5063   × cxp 5552  1-1-ontowf1o 6353  cfv 6354   Isom wiso 6355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-xp 5560  df-iota 6313  df-fv 6362  df-isom 6363
This theorem is referenced by: (None)
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