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Theorem ersym 6250
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 6247 . . . . . 6  |-  ( R  Er  X  ->  Rel  R )
42, 3syl 14 . . . . 5  |-  ( ph  ->  Rel  R )
5 brrelex12 4446 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
64, 1, 5syl2anc 403 . . . 4  |-  ( ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
7 brcnvg 4584 . . . . 5  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B `' R A 
<->  A R B ) )
87ancoms 264 . . . 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 165 . 2  |-  ( ph  ->  B `' R A )
11 df-er 6238 . . . . . 6  |-  ( R  Er  X  <->  ( Rel  R  /\  dom  R  =  X  /\  ( `' R  u.  ( R  o.  R ) ) 
C_  R ) )
1211simp3bi 958 . . . . 5  |-  ( R  Er  X  ->  ( `' R  u.  ( R  o.  R )
)  C_  R )
132, 12syl 14 . . . 4  |-  ( ph  ->  ( `' R  u.  ( R  o.  R
) )  C_  R
)
1413unssad 3166 . . 3  |-  ( ph  ->  `' R  C_  R )
1514ssbrd 3861 . 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 102    <-> wb 103    = wceq 1287    e. wcel 1436   _Vcvv 2615    u. cun 2986    C_ wss 2988   class class class wbr 3820   `'ccnv 4409   dom cdm 4410    o. ccom 4414   Rel wrel 4415    Er wer 6235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-pow 3983  ax-pr 4009
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-xp 4416  df-rel 4417  df-cnv 4418  df-er 6238
This theorem is referenced by:  ercl2  6251  ersymb  6252  ertr2d  6255  ertr3d  6256  ertr4d  6257  erth  6282  erinxp  6312
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