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Theorem erref 6680
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 )

Proof of Theorem erref
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 erref.2 . . . 4  |-  ( ph  ->  A  e.  X )
2 ersymb.1 . . . . 5  |-  ( ph  ->  R  Er  X )
3 erdm 6670 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 15 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2360 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 4874 . . . 4  |-  ( A  e.  X  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
71, 6syl 15 . . 3  |-  ( ph  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
85, 7mpbid 201 . 2  |-  ( ph  ->  E. x  A R x )
92adantr 451 . . . . 5  |-  ( (
ph  /\  A R x )  ->  R  Er  X )
10 simpr 447 . . . . 5  |-  ( (
ph  /\  A R x )  ->  A R x )
119, 10, 10ertr4d 6679 . . . 4  |-  ( (
ph  /\  A R x )  ->  A R A )
1211ex 423 . . 3  |-  ( ph  ->  ( A R x  ->  A R A ) )
1312exlimdv 1664 . 2  |-  ( ph  ->  ( E. x  A R x  ->  A R A ) )
148, 13mpd 14 1  |-  ( ph  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   class class class wbr 4023   dom cdm 4689    Er wer 6657
This theorem is referenced by:  iserd  6686  erth  6704  iiner  6731  erinxp  6733  nqerid  8557  enqeq  8558  divsgrp  14672  sylow2alem1  14928  sylow2alem2  14929  sylow2a  14930  efginvrel2  15036  efgsrel  15043  efgcpbllemb  15064  frgp0  15069  frgpnabllem1  15161  frgpnabllem2  15162  pcophtb  18527  pi1xfrf  18551  pi1xfr  18553  pi1xfrcnvlem  18554  prtlem10  26733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-er 6660
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