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Theorem isof1oopb 6615
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 6239 . . . . . . . . 9 (𝐻𝑥) ∈ V
2 fvex 6239 . . . . . . . . 9 (𝐻𝑦) ∈ V
31, 2opelvv 5200 . . . . . . . 8 ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V)
4 df-br 4686 . . . . . . . 8 ((𝐻𝑥)(V × V)(𝐻𝑦) ↔ ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V))
53, 4mpbir 221 . . . . . . 7 (𝐻𝑥)(V × V)(𝐻𝑦)
65a1i 11 . . . . . 6 (𝑥(V × V)𝑦 → (𝐻𝑥)(V × V)(𝐻𝑦))
7 opelvvg 5199 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V))
8 df-br 4686 . . . . . . . 8 (𝑥(V × V)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V))
97, 8sylibr 224 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → 𝑥(V × V)𝑦)
109a1d 25 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ((𝐻𝑥)(V × V)(𝐻𝑦) → 𝑥(V × V)𝑦))
116, 10impbid2 216 . . . . 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 3000 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1413pm4.71i 665 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
15 df-isom 5935 . 2 (𝐻 Isom (V × V), (V × V)(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
1614, 15bitr4i 267 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030  wral 2941  Vcvv 3231  cop 4216   class class class wbr 4685   × cxp 5141  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-iota 5889  df-fv 5934  df-isom 5935
This theorem is referenced by: (None)
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