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

Theorem brcnvg 4861
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  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )

Proof of Theorem brcnvg
StepHypRef Expression
1 opelcnvg 4860 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
2 df-br 4025 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
3 df-br 4025 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
41, 2, 33bitr4g 281 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1685   <.cop 3644   class class class wbr 4024   `'ccnv 4687
This theorem is referenced by:  brcnv  4863  brelrng  4907  eliniseg  5041  relbrcnvg  5051  brcodir  5061  dffv2  5553  ersym  6667  brdifun  6682  lbinfm  9702  infmrgelb  9729  infmrlb  9730  infmxrlb  10646  infmxrgelb  10647  oduleg  14230  posglbd  14247  znleval  16502  ballotlemirc  23083  inffz  23498  elpredg  23579  predep  23593  brbtwn  23934  colineardim1  24091  cnvref  24463  mnlmxl2  24668  nfwpr4c  24684  toplat  24689  gtinf  25633  infrglb  27121  gte-lte  27454  gt-lt  27455
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-cnv 4696
  Copyright terms: Public domain W3C validator