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Theorem ersym 6920
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 6917 . . . . . 6  |-  ( R  Er  X  ->  Rel  R )
42, 3syl 16 . . . . 5  |-  ( ph  ->  Rel  R )
5 brrelex12 4918 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
64, 1, 5syl2anc 644 . . . 4  |-  ( ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
7 brcnvg 5056 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B `' R A 
<->  A R B ) )
87ancoms 441 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( B `' R A 
<->  A R B ) )
96, 8syl 16 . . 3  |-  ( ph  ->  ( B `' R A 
<->  A R B ) )
101, 9mpbird 225 . 2  |-  ( ph  ->  B `' R A )
11 df-er 6908 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
1211simp3bi 975 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
132, 12syl 16 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
1413unssad 3526 . . 3  |-  ( ph  ->  `' R  C_  R )
1514ssbrd 4256 . 2  |-  ( ph  ->  ( B `' R A  ->  B R A ) )
1610, 15mpd 15 1  |-  ( ph  ->  B R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320    C_ wss 3322   class class class wbr 4215   `'ccnv 4880   dom cdm 4881    o. ccom 4885   Rel wrel 4886    Er wer 6905
This theorem is referenced by:  ercl2  6921  ersymb  6922  ertr2d  6925  ertr3d  6926  ertr4d  6927  erth  6952  erinxp  6981  nqereu  8811  nqerf  8812  1nqenq  8844  divsgrp2  14941  efginvrel2  15364  efgcpbllemb  15392  2idlcpbl  16310  tgptsmscls  18184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-er 6908
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