Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isof1oopb Structured version   Visualization version   GIF version

Theorem isof1oopb 6475
 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 6113 . . . . . . . . 9 (𝐻𝑥) ∈ V
2 fvex 6113 . . . . . . . . 9 (𝐻𝑦) ∈ V
31, 2opelvv 5088 . . . . . . . 8 ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V)
4 df-br 4584 . . . . . . . 8 ((𝐻𝑥)(V × V)(𝐻𝑦) ↔ ⟨(𝐻𝑥), (𝐻𝑦)⟩ ∈ (V × V))
53, 4mpbir 220 . . . . . . 7 (𝐻𝑥)(V × V)(𝐻𝑦)
65a1i 11 . . . . . 6 (𝑥(V × V)𝑦 → (𝐻𝑥)(V × V)(𝐻𝑦))
7 opelvvg 5087 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × V))
8 df-br 4584 . . . . . . . 8 (𝑥(V × V)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V))
97, 8sylibr 223 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → 𝑥(V × V)𝑦)
109a1d 25 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ((𝐻𝑥)(V × V)(𝐻𝑦) → 𝑥(V × V)𝑦))
116, 10impbid2 215 . . . . 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 2954 . . 3 (𝐻:𝐴1-1-onto𝐵 → ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦)))
1413pm4.71i 662 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
15 df-isom 5813 . 2 (𝐻 Isom (V × V), (V × V)(𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(V × V)𝑦 ↔ (𝐻𝑥)(V × V)(𝐻𝑦))))
1614, 15bitr4i 266 1 (𝐻:𝐴1-1-onto𝐵𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   × cxp 5036  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-iota 5768  df-fv 5812  df-isom 5813 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator