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Theorem relbrcnvg 5052
Description: When  R is a relation, the sethood assumptions on brcnv 4864 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 5051 . . . 4  |-  Rel  `' R
2 brrelex12 4726 . . . 4  |-  ( ( Rel  `' R  /\  A `' R B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
31, 2mpan 651 . . 3  |-  ( A `' R B  ->  ( A  e.  _V  /\  B  e.  _V ) )
43a1i 10 . 2  |-  ( Rel 
R  ->  ( A `' R B  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
5 brrelex12 4726 . . . 4  |-  ( ( Rel  R  /\  B R A )  ->  ( B  e.  _V  /\  A  e.  _V ) )
65ancomd 438 . . 3  |-  ( ( Rel  R  /\  B R A )  ->  ( A  e.  _V  /\  B  e.  _V ) )
76ex 423 . 2  |-  ( Rel 
R  ->  ( B R A  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
8 brcnvg 4862 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A `' R B 
<->  B R A ) )
98a1i 10 . 2  |-  ( Rel 
R  ->  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A `' R B  <->  B R A ) ) )
104, 7, 9pm5.21ndd 343 1  |-  ( Rel 
R  ->  ( A `' R B  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   `'ccnv 4688   Rel wrel 4694
This theorem is referenced by:  eliniseg2  5053  relbrcnv  5054  isinv  13662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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