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Theorem relbrcnvg 4888
Description: When  R is a relation, the sethood assumptions on brcnv 4692 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 4887 . . . 4  |-  Rel  `' R
2 brrelex12 4547 . . . 4  |-  ( ( Rel  `' R  /\  A `' R B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
31, 2mpan 420 . . 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 4547 . . . 4  |-  ( ( Rel  R  /\  B R A )  ->  ( B  e.  _V  /\  A  e.  _V ) )
65ancomd 265 . . 3  |-  ( ( Rel  R  /\  B R A )  ->  ( A  e.  _V  /\  B  e.  _V ) )
76ex 114 . 2  |-  ( Rel 
R  ->  ( B R A  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
8 brcnvg 4690 . . 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 679 1  |-  ( Rel 
R  ->  ( A `' R B  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   _Vcvv 2660   class class class wbr 3899   `'ccnv 4508   Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by:  relbrcnv  4889
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