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Theorem brcnvg 5263
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 5262 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
2 df-br 4614 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3 df-br 4614 . 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 4154   class class class wbr 4613  ccnv 5073
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 4741  ax-nul 4749  ax-pr 4867
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 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-cnv 5082
This theorem is referenced by:  brcnv  5265  brelrng  5315  eliniseg  5453  relbrcnvg  5463  brcodir  5474  elpredg  5653  predep  5665  dffv2  6228  ersym  7699  brdifun  7716  eqinf  8334  inflb  8339  infglb  8340  infglbb  8341  infltoreq  8352  infempty  8356  lbinf  10920  brcnvtrclfv  13678  oduleg  17053  posglbd  17071  znleval  19822  brbtwn  25679  fcoinvbr  29262  cnvordtrestixx  29741  xrge0iifiso  29763  orvcgteel  30310  inffzOLD  31323  fv1stcnv  31382  fv2ndcnv  31383  wsuclem  31474  wsuclemOLD  31475  wsuclb  31478  sltgtres  31570  noextendltgt  31574  colineardim1  31810  gtinfOLD  31956  brnonrel  37376  ntrneifv2  37860  gte-lte  41758  gt-lt  41759
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