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Theorem erref 6884
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 6874 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 16 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2481 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 5024 . . . 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 6883 . 2  |-  ( (
ph  /\  A R x )  ->  A R A )
128, 11exlimddv 1645 1  |-  ( ph  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   class class class wbr 4172   dom cdm 4837    Er wer 6861
This theorem is referenced by:  iserd  6890  erth  6908  iiner  6935  erinxp  6937  nqerid  8766  enqeq  8767  divsgrp  14950  sylow2alem1  15206  sylow2alem2  15207  sylow2a  15208  efginvrel2  15314  efgsrel  15321  efgcpbllemb  15342  frgp0  15347  frgpnabllem1  15439  frgpnabllem2  15440  pcophtb  19007  pi1xfrf  19031  pi1xfr  19033  pi1xfrcnvlem  19034  prtlem10  26604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-er 6864
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