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Theorem erref 6192
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 6182 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 14 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2159 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 4558 . . . 4  |-  ( A  e.  X  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
71, 6syl 14 . . 3  |-  ( ph  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
85, 7mpbid 145 . 2  |-  ( ph  ->  E. x  A R x )
92adantr 270 . . 3  |-  ( (
ph  /\  A R x )  ->  R  Er  X )
10 simpr 108 . . 3  |-  ( (
ph  /\  A R x )  ->  A R x )
119, 10, 10ertr4d 6191 . 2  |-  ( (
ph  /\  A R x )  ->  A R A )
128, 11exlimddv 1820 1  |-  ( ph  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   class class class wbr 3793   dom cdm 4371    Er wer 6169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-er 6172
This theorem is referenced by:  iserd  6198  erth  6216  iinerm  6244  erinxp  6246
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