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Theorem opelcnvg 5440
Description: Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
opelcnvg ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))

Proof of Theorem opelcnvg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4790 . . 3 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
2 breq1 4789 . . 3 (𝑦 = 𝐵 → (𝑦𝑅𝐴𝐵𝑅𝐴))
3 df-cnv 5257 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
41, 2, 3brabg 5127 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))
5 df-br 4787 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
6 df-br 4787 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
74, 5, 63bitr3g 302 1 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  cop 4322   class class class wbr 4786  ccnv 5248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-cnv 5257
This theorem is referenced by:  brcnvg  5441  opelcnv  5442  elpredim  5835  fvimacnv  6475  brtpos  7513  xrlenlt  10305  brcolinear2  32502
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