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Theorem erref 6292
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 6282 . . . . 5  |-  ( R  Er  X  ->  dom  R  =  X )
42, 3syl 14 . . . 4  |-  ( ph  ->  dom  R  =  X )
51, 4eleqtrrd 2167 . . 3  |-  ( ph  ->  A  e.  dom  R
)
6 eldmg 4619 . . . 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 6291 . 2  |-  ( (
ph  /\  A R x )  ->  A R A )
128, 11exlimddv 1826 1  |-  ( ph  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   class class class wbr 3837   dom cdm 4428    Er wer 6269
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-er 6272
This theorem is referenced by:  iserd  6298  erth  6316  iinerm  6344  erinxp  6346
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