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Theorem erref 6634
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 6624 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 17 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2333 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 4848 . . . 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 6633 . . . 4  |-  ( (
ph  /\  A R x )  ->  A R A )
1211ex 425 . . 3  |-  ( ph  ->  ( A R x  ->  A R A ) )
1312exlimdv 1933 . 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 3983   dom cdm 4647    Er wer 6611
This theorem is referenced by:  iserd  6640  erth  6658  iiner  6685  erinxp  6687  nqerid  8511  enqeq  8512  divsgrp  14620  sylow2alem1  14876  sylow2alem2  14877  sylow2a  14878  efginvrel2  14984  efgsrel  14991  efgcpbllemb  15012  frgp0  15017  frgpnabllem1  15109  frgpnabllem2  15110  pcophtb  18475  pi1xfrf  18499  pi1xfr  18501  pi1xfrcnvlem  18502  prtlem10  26086
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-er 6614
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