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

Theorem brcnvg 5441
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 5440 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
2 df-br 4787 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3 df-br 4787 . 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 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:  brcnv  5443  brelrng  5493  eliniseg  5635  relbrcnvg  5645  brcodir  5656  elpredg  5837  predep  5849  dffv2  6413  ersym  7908  brdifun  7925  eqinf  8546  inflb  8551  infglb  8552  infglbb  8553  infltoreq  8564  infempty  8568  brcnvtrclfv  13952  oduleg  17340  posglbd  17358  znleval  20118  brbtwn  26000  fcoinvbr  29757  cnvordtrestixx  30299  xrge0iifiso  30321  orvcgteel  30869  inffzOLD  31953  fv1stcnv  32016  fv2ndcnv  32017  wsuclem  32107  wsuclb  32110  slenlt  32214  colineardim1  32505  gtinfOLD  32651  eldmcnv  34455  ineccnvmo  34464  alrmomorn  34465  brxrn  34478  dfcoss3  34514  brcoss3  34530  brcosscnv  34564  cosscnvssid3  34568  cosscnvssid4  34569  brnonrel  38421  ntrneifv2  38904  gte-lte  42996  gt-lt  42997
  Copyright terms: Public domain W3C validator