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Theorem erref 6696
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 6686 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 15 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2373 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 4890 . . . 4  |-  ( A  e.  X  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
71, 6syl 15 . . 3  |-  ( ph  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
85, 7mpbid 201 . 2  |-  ( ph  ->  E. x  A R x )
92adantr 451 . . . . 5  |-  ( (
ph  /\  A R x )  ->  R  Er  X )
10 simpr 447 . . . . 5  |-  ( (
ph  /\  A R x )  ->  A R x )
119, 10, 10ertr4d 6695 . . . 4  |-  ( (
ph  /\  A R x )  ->  A R A )
1211ex 423 . . 3  |-  ( ph  ->  ( A R x  ->  A R A ) )
1312exlimdv 1626 . 2  |-  ( ph  ->  ( E. x  A R x  ->  A R A ) )
148, 13mpd 14 1  |-  ( ph  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705    Er wer 6673
This theorem is referenced by:  iserd  6702  erth  6720  iiner  6747  erinxp  6749  nqerid  8573  enqeq  8574  divsgrp  14688  sylow2alem1  14944  sylow2alem2  14945  sylow2a  14946  efginvrel2  15052  efgsrel  15059  efgcpbllemb  15080  frgp0  15085  frgpnabllem1  15177  frgpnabllem2  15178  pcophtb  18543  pi1xfrf  18567  pi1xfr  18569  pi1xfrcnvlem  18570  prtlem10  26836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-er 6676
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