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Theorem ersym 6814
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 6811 . . . . . 6  |-  ( R  Er  X  ->  Rel  R )
42, 3syl 15 . . . . 5  |-  ( ph  ->  Rel  R )
5 brrelex12 4829 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
64, 1, 5syl2anc 642 . . . 4  |-  ( ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
7 brcnvg 4965 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B `' R A 
<->  A R B ) )
87ancoms 439 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B `' R A 
<->  A R B ) )
96, 8syl 15 . . 3  |-  ( ph  ->  ( B `' R A 
<->  A R B ) )
101, 9mpbird 223 . 2  |-  ( ph  ->  B `' R A )
11 ssun1 3426 . . . 4  |-  `' R  C_  ( `' R  u.  ( R  o.  R
) )
12 df-er 6802 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
1312simp3bi 973 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
142, 13syl 15 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
1511, 14syl5ss 3276 . . 3  |-  ( ph  ->  `' R  C_  R )
1615ssbrd 4166 . 2  |-  ( ph  ->  ( B `' R A  ->  B R A ) )
1710, 16mpd 14 1  |-  ( ph  ->  B R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873    u. cun 3236    C_ wss 3238   class class class wbr 4125   `'ccnv 4791   dom cdm 4792    o. ccom 4796   Rel wrel 4797    Er wer 6799
This theorem is referenced by:  ercl2  6815  ersymb  6816  ertr2d  6819  ertr3d  6820  ertr4d  6821  erth  6846  erinxp  6875  nqereu  8700  nqerf  8701  1nqenq  8733  divsgrp2  14823  efginvrel2  15246  efgcpbllemb  15274  2idlcpbl  16196  tgptsmscls  18045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-cnv 4800  df-er 6802
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