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Theorem ersym 6409
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 6406 . . . . . 6  |-  ( R  Er  X  ->  Rel  R )
42, 3syl 14 . . . . 5  |-  ( ph  ->  Rel  R )
5 brrelex12 4547 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
64, 1, 5syl2anc 408 . . . 4  |-  ( ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
7 brcnvg 4690 . . . . 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 6397 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
1211simp3bi 983 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
132, 12syl 14 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
1413unssad 3223 . . 3  |-  ( ph  ->  `' R  C_  R )
1514ssbrd 3941 . 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 1316    e. wcel 1465   _Vcvv 2660    u. cun 3039    C_ wss 3041   class class class wbr 3899   `'ccnv 4508   dom cdm 4509    o. ccom 4513   Rel wrel 4514    Er wer 6394
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-er 6397
This theorem is referenced by:  ercl2  6410  ersymb  6411  ertr2d  6414  ertr3d  6415  ertr4d  6416  erth  6441  erinxp  6471
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