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Theorem relbrcnvg 4764
Description: When  R is a relation, the sethood assumptions on brcnv 4575 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 4763 . . . 4  |-  Rel  `' R
2 brrelex12 4435 . . . 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 4435 . . . 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 4573 . . 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 654 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 1434   _Vcvv 2612   class class class wbr 3811   `'ccnv 4398   Rel wrel 4404
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4405  df-rel 4406  df-cnv 4407
This theorem is referenced by:  relbrcnv  4765
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