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Theorem relbrcnvg 4811
Description: When  R is a relation, the sethood assumptions on brcnv 4619 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
relbrcnvg  |-  ( Rel 
R  ->  ( A `' R B  <->  B R A ) )

Proof of Theorem relbrcnvg
StepHypRef Expression
1 relcnv 4810 . . . 4  |-  Rel  `' R
2 brrelex12 4475 . . . 4  |-  ( ( Rel  `' R  /\  A `' R B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
31, 2mpan 415 . . 3  |-  ( A `' R B  ->  ( A  e.  _V  /\  B  e.  _V ) )
43a1i 9 . 2  |-  ( Rel 
R  ->  ( A `' R B  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
5 brrelex12 4475 . . . 4  |-  ( ( Rel  R  /\  B R A )  ->  ( B  e.  _V  /\  A  e.  _V ) )
65ancomd 263 . . 3  |-  ( ( Rel  R  /\  B R A )  ->  ( A  e.  _V  /\  B  e.  _V ) )
76ex 113 . 2  |-  ( Rel 
R  ->  ( B R A  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
8 brcnvg 4617 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A `' R B 
<->  B R A ) )
98a1i 9 . 2  |-  ( Rel 
R  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A `' R B  <->  B R A ) ) )
104, 7, 9pm5.21ndd 656 1  |-  ( Rel 
R  ->  ( A `' R B  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438   _Vcvv 2619   class class class wbr 3845   `'ccnv 4437   Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446
This theorem is referenced by:  relbrcnv  4812
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