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Theorem isof1oopb 6739
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 6363 . . . . . . . . 9 (𝐻𝑥) ∈ V
2 fvex 6363 . . . . . . . . 9 (𝐻𝑦) ∈ V
31, 2opelvv 5323 . . . . . . . 8 ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V)
4 df-br 4805 . . . . . . . 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 5322 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V))
8 df-br 4805 . . . . . . . 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 473 . . . 4 ((𝐻:𝐴1-1-onto𝐵 ∧ (𝑥𝐴𝑦𝐴)) → (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1312ralrimivva 3109 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1413pm4.71i 667 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
15 df-isom 6058 . 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 2139  wral 3050  Vcvv 3340  cop 4327   class class class wbr 4804   × cxp 5264  1-1-ontowf1o 6048  cfv 6049   Isom wiso 6050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-iota 6012  df-fv 6057  df-isom 6058
This theorem is referenced by: (None)
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