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Theorem erref 7328
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 7318 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 16 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2558 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 5196 . . . 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 7327 . 2  |-  ( (
ph  /\  A R x )  ->  A R A )
128, 11exlimddv 1702 1  |-  ( ph  ->  A R A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   class class class wbr 4447   dom cdm 4999    Er wer 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-er 7308
This theorem is referenced by:  iserd  7334  erth  7353  iiner  7380  erinxp  7382  nqerid  9307  enqeq  9308  divsgrp  16051  sylow2alem1  16433  sylow2alem2  16434  sylow2a  16435  efginvrel2  16541  efgsrel  16548  efgcpbllemb  16569  frgp0  16574  frgpnabllem1  16668  frgpnabllem2  16669  pcophtb  21264  pi1xfrf  21288  pi1xfr  21290  pi1xfrcnvlem  21291  prtlem10  30210
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