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Theorem relbrcnvg 5184
Description: When  R is a relation, the sethood assumptions on brcnv 4996 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 5183 . . . 4  |-  Rel  `' R
2 brrelex12 4856 . . . 4  |-  ( ( Rel  `' R  /\  A `' R B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
31, 2mpan 652 . . 3  |-  ( A `' R B  ->  ( A  e.  _V  /\  B  e.  _V ) )
43a1i 11 . 2  |-  ( Rel 
R  ->  ( A `' R B  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
5 brrelex12 4856 . . . 4  |-  ( ( Rel  R  /\  B R A )  ->  ( B  e.  _V  /\  A  e.  _V ) )
65ancomd 439 . . 3  |-  ( ( Rel  R  /\  B R A )  ->  ( A  e.  _V  /\  B  e.  _V ) )
76ex 424 . 2  |-  ( Rel 
R  ->  ( B R A  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
8 brcnvg 4994 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A `' R B 
<->  B R A ) )
98a1i 11 . 2  |-  ( Rel 
R  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A `' R B  <->  B R A ) ) )
104, 7, 9pm5.21ndd 344 1  |-  ( Rel 
R  ->  ( A `' R B  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   _Vcvv 2900   class class class wbr 4154   `'ccnv 4818   Rel wrel 4824
This theorem is referenced by:  eliniseg2  5185  relbrcnv  5186  isinv  13913
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827
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