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Theorem erref 7126
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 7116 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 16 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2520 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 5040 . . . 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 210 . 2  |-  ( ph  ->  E. x  A R x )
92adantr 465 . . 3  |-  ( (
ph  /\  A R x )  ->  R  Er  X )
10 simpr 461 . . 3  |-  ( (
ph  /\  A R x )  ->  A R x )
119, 10, 10ertr4d 7125 . 2  |-  ( (
ph  /\  A R x )  ->  A R A )
128, 11exlimddv 1692 1  |-  ( ph  ->  A R A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   class class class wbr 4297   dom cdm 4845    Er wer 7103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-er 7106
This theorem is referenced by:  iserd  7132  erth  7150  iiner  7177  erinxp  7179  nqerid  9107  enqeq  9108  divsgrp  15741  sylow2alem1  16121  sylow2alem2  16122  sylow2a  16123  efginvrel2  16229  efgsrel  16236  efgcpbllemb  16257  frgp0  16262  frgpnabllem1  16356  frgpnabllem2  16357  pcophtb  20606  pi1xfrf  20630  pi1xfr  20632  pi1xfrcnvlem  20633  prtlem10  29015
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