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Theorem brcnvg 5012
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 5011 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
2 df-br 4173 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
3 df-br 4173 . 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 1721   <.cop 3777   class class class wbr 4172   `'ccnv 4836
This theorem is referenced by:  brcnv  5014  brelrng  5058  eliniseg  5192  relbrcnvg  5202  brcodir  5212  dffv2  5755  ersym  6876  brdifun  6891  lbinfm  9917  infmrgelb  9944  infmrlb  9945  infmxrlb  10868  infmxrgelb  10869  oduleg  14514  posglbd  14531  znleval  16790  tosglb  24145  cnvordtrestixx  24264  xrge0iifiso  24274  orvcgteel  24678  ballotlemirc  24742  inffz  25153  elpredg  25392  predep  25406  brbtwn  25742  colineardim1  25899  gtinf  26212  infrglb  27589  gte-lte  28181  gt-lt  28182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cnv 4845
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