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Theorem erref 6566
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
StepHypRef Expression
1 erref.2 . . . 4  |-  ( ph  ->  A  e.  X )
2 ersymb.1 . . . . 5  |-  ( ph  ->  R  Er  X )
3 erdm 6556 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 17 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2330 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 4781 . . . 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 6565 . . . 4  |-  ( (
ph  /\  A R x )  ->  A R A )
1211ex 425 . . 3  |-  ( ph  ->  ( A R x  ->  A R A ) )
1312exlimdv 1932 . 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 1537    = wceq 1619    e. wcel 1621   class class class wbr 3920   dom cdm 4580    Er wer 6543
This theorem is referenced by:  iserd  6572  erth  6590  iiner  6617  erinxp  6619  nqerid  8437  enqeq  8438  divsgrp  14507  sylow2alem1  14763  sylow2alem2  14764  sylow2a  14765  efginvrel2  14871  efgsrel  14878  efgcpbllemb  14899  frgp0  14904  frgpnabllem1  14996  frgpnabllem2  14997  pcophtb  18359  pi1xfrf  18383  pi1xfr  18385  pi1xfrcnvlem  18386  prtlem10  25899
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-er 6546
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