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Theorem erref 6675
Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersymb.1  |-  ( ph  ->  R  Er  X )
erref.2  |-  ( ph  ->  A  e.  X )
Assertion
Ref Expression
erref  |-  ( ph  ->  A R A )
Dummy variable  x is distinct from all other variables.

Proof of Theorem erref
StepHypRef Expression
1 erref.2 . . . 4  |-  ( ph  ->  A  e.  X )
2 ersymb.1 . . . . 5  |-  ( ph  ->  R  Er  X )
3 erdm 6665 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 17 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2361 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 4873 . . . 4  |-  ( A  e.  X  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
71, 6syl 17 . . 3  |-  ( ph  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
85, 7mpbid 203 . 2  |-  ( ph  ->  E. x  A R x )
92adantr 453 . . . . 5  |-  ( (
ph  /\  A R x )  ->  R  Er  X )
10 simpr 449 . . . . 5  |-  ( (
ph  /\  A R x )  ->  A R x )
119, 10, 10ertr4d 6674 . . . 4  |-  ( (
ph  /\  A R x )  ->  A R A )
1211ex 425 . . 3  |-  ( ph  ->  ( A R x  ->  A R A ) )
1312exlimdv 1665 . 2  |-  ( ph  ->  ( E. x  A R x  ->  A R A ) )
148, 13mpd 16 1  |-  ( ph  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   class class class wbr 4024   dom cdm 4688    Er wer 6652
This theorem is referenced by:  iserd  6681  erth  6699  iiner  6726  erinxp  6728  nqerid  8552  enqeq  8553  divsgrp  14666  sylow2alem1  14922  sylow2alem2  14923  sylow2a  14924  efginvrel2  15030  efgsrel  15037  efgcpbllemb  15058  frgp0  15063  frgpnabllem1  15155  frgpnabllem2  15156  pcophtb  18521  pi1xfrf  18545  pi1xfr  18547  pi1xfrcnvlem  18548  prtlem10  26132
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-er 6655
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