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Theorem erref 7368
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 7358 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 17 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2493 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 5019 . . . 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 210 . 2  |-  ( ph  ->  E. x  A R x )
92adantr 463 . . 3  |-  ( (
ph  /\  A R x )  ->  R  Er  X )
10 simpr 459 . . 3  |-  ( (
ph  /\  A R x )  ->  A R x )
119, 10, 10ertr4d 7367 . 2  |-  ( (
ph  /\  A R x )  ->  A R A )
128, 11exlimddv 1747 1  |-  ( ph  ->  A R A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   class class class wbr 4395   dom cdm 4823    Er wer 7345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-er 7348
This theorem is referenced by:  iserd  7374  erth  7393  iiner  7420  erinxp  7422  nqerid  9341  enqeq  9342  qusgrp  16580  sylow2alem1  16961  sylow2alem2  16962  sylow2a  16963  efginvrel2  17069  efgsrel  17076  efgcpbllemb  17097  frgp0  17102  frgpnabllem1  17201  frgpnabllem2  17202  pcophtb  21821  pi1xfrf  21845  pi1xfr  21847  pi1xfrcnvlem  21848  prtlem10  31888
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