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Theorem erref 6916
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 6906 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 16 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2512 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 5056 . . . 4  |-  ( A  e.  X  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
71, 6syl 16 . . 3  |-  ( ph  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
85, 7mpbid 202 . 2  |-  ( ph  ->  E. x  A R x )
92adantr 452 . . 3  |-  ( (
ph  /\  A R x )  ->  R  Er  X )
10 simpr 448 . . 3  |-  ( (
ph  /\  A R x )  ->  A R x )
119, 10, 10ertr4d 6915 . 2  |-  ( (
ph  /\  A R x )  ->  A R A )
128, 11exlimddv 1648 1  |-  ( ph  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   class class class wbr 4204   dom cdm 4869    Er wer 6893
This theorem is referenced by:  iserd  6922  erth  6940  iiner  6967  erinxp  6969  nqerid  8799  enqeq  8800  divsgrp  14983  sylow2alem1  15239  sylow2alem2  15240  sylow2a  15241  efginvrel2  15347  efgsrel  15354  efgcpbllemb  15375  frgp0  15380  frgpnabllem1  15472  frgpnabllem2  15473  pcophtb  19042  pi1xfrf  19066  pi1xfr  19068  pi1xfrcnvlem  19069  prtlem10  26651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-er 6896
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