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Theorem brcnvg 5053
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 5052 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
2 df-br 4213 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
3 df-br 4213 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
41, 2, 33bitr4g 280 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   <.cop 3817   class class class wbr 4212   `'ccnv 4877
This theorem is referenced by:  brcnv  5055  brelrng  5099  eliniseg  5233  relbrcnvg  5243  brcodir  5253  dffv2  5796  ersym  6917  brdifun  6932  lbinfm  9961  infmrgelb  9988  infmrlb  9989  infmxrlb  10912  infmxrgelb  10913  oduleg  14559  posglbd  14576  znleval  16835  tosglb  24192  cnvordtrestixx  24311  xrge0iifiso  24321  orvcgteel  24725  ballotlemirc  24789  inffz  25200  elpredg  25453  predep  25467  wsuclem  25576  wsuclb  25579  brbtwn  25838  colineardim1  25995  gtinf  26322  infrglb  27698  gte-lte  28467  gt-lt  28468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886
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