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Theorem brcnvg 5273
Description: The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
Assertion
Ref Expression
brcnvg ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem brcnvg
StepHypRef Expression
1 opelcnvg 5272 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
2 df-br 4624 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3 df-br 4624 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
41, 2, 33bitr4g 303 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  cop 4161   class class class wbr 4623  ccnv 5083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-cnv 5092
This theorem is referenced by:  brcnv  5275  brelrng  5325  eliniseg  5463  relbrcnvg  5473  brcodir  5484  elpredg  5663  predep  5675  dffv2  6238  ersym  7714  brdifun  7731  eqinf  8350  inflb  8355  infglb  8356  infglbb  8357  infltoreq  8368  infempty  8372  lbinf  10936  brcnvtrclfv  13694  oduleg  17072  posglbd  17090  znleval  19843  brbtwn  25713  fcoinvbr  29303  cnvordtrestixx  29783  xrge0iifiso  29805  orvcgteel  30352  inffzOLD  31376  fv1stcnv  31435  fv2ndcnv  31436  wsuclem  31527  wsuclemOLD  31528  wsuclb  31531  sltgtres  31623  noextendltgt  31627  colineardim1  31863  gtinfOLD  32009  brnonrel  37415  ntrneifv2  37899  gte-lte  41788  gt-lt  41789
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