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Theorem ersym 6407
Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1  |-  ( ph  ->  R  Er  X )
ersym.2  |-  ( ph  ->  A R B )
Assertion
Ref Expression
ersym  |-  ( ph  ->  B R A )

Proof of Theorem ersym
StepHypRef Expression
1 ersym.2 . . 3  |-  ( ph  ->  A R B )
2 ersym.1 . . . . . 6  |-  ( ph  ->  R  Er  X )
3 errel 6404 . . . . . 6  |-  ( R  Er  X  ->  Rel  R )
42, 3syl 14 . . . . 5  |-  ( ph  ->  Rel  R )
5 brrelex12 4545 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
64, 1, 5syl2anc 406 . . . 4  |-  ( ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
7 brcnvg 4688 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B `' R A 
<->  A R B ) )
87ancoms 266 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B `' R A 
<->  A R B ) )
96, 8syl 14 . . 3  |-  ( ph  ->  ( B `' R A 
<->  A R B ) )
101, 9mpbird 166 . 2  |-  ( ph  ->  B `' R A )
11 df-er 6395 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
1211simp3bi 981 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
132, 12syl 14 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
1413unssad 3221 . . 3  |-  ( ph  ->  `' R  C_  R )
1514ssbrd 3939 . 2  |-  ( ph  ->  ( B `' R A  ->  B R A ) )
1610, 15mpd 13 1  |-  ( ph  ->  B R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   _Vcvv 2658    u. cun 3037    C_ wss 3039   class class class wbr 3897   `'ccnv 4506   dom cdm 4507    o. ccom 4511   Rel wrel 4512    Er wer 6392
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-er 6395
This theorem is referenced by:  ercl2  6408  ersymb  6409  ertr2d  6412  ertr3d  6413  ertr4d  6414  erth  6439  erinxp  6469
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