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Theorem brcnvg 4807
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 4806 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
2 df-br 4003 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
3 df-br 4003 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
41, 2, 33bitr4g 223 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2148   <.cop 3595   class class class wbr 4002   `'ccnv 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-cnv 4633
This theorem is referenced by:  brcnv  4809  brelrng  4857  eliniseg  4997  relbrcnvg  5006  brcodir  5015  sefvex  5535  foeqcnvco  5788  isocnv2  5810  ersym  6544  brdifun  6559  ecidg  6596  cnvti  7015  eqinfti  7016  inflbti  7020  infglbti  7021  negiso  8908  xrnegiso  11263  pw1nct  14612
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