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

Theorem brcnvg 5225
 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 5224 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
2 df-br 4584 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3 df-br 4584 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
41, 2, 33bitr4g 302 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037 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-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  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-br 4584  df-opab 4644  df-cnv 5046 This theorem is referenced by:  brcnv  5227  brelrng  5276  eliniseg  5413  relbrcnvg  5423  brcodir  5434  elpredg  5611  predep  5623  dffv2  6181  ersym  7641  brdifun  7658  eqinf  8273  inflb  8278  infglb  8279  infglbb  8280  infltoreq  8291  infempty  8295  lbinf  10855  brcnvtrclfv  13592  oduleg  16955  posglbd  16973  znleval  19722  brbtwn  25579  fcoinvbr  28799  cnvordtrestixx  29287  xrge0iifiso  29309  orvcgteel  29856  inffzOLD  30868  fv1stcnv  30925  fv2ndcnv  30926  wsuclem  31017  wsuclemOLD  31018  wsuclb  31021  colineardim1  31338  gtinfOLD  31484  brnonrel  36914  ntrneifv2  37398  gte-lte  42264  gt-lt  42265
 Copyright terms: Public domain W3C validator