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Theorem erref 6062
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  R  Er  X
erref.2  X
Assertion
Ref Expression
erref  R

Proof of Theorem erref
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 erref.2 . . . 4  X
2 ersymb.1 . . . . 5  R  Er  X
3 erdm 6052 . . . . 5  R  Er  X  dom  R  X
42, 3syl 14 . . . 4  dom  R  X
51, 4eleqtrrd 2114 . . 3  dom  R
6 eldmg 4473 . . . 4  X  dom  R  R
71, 6syl 14 . . 3  dom  R  R
85, 7mpbid 135 . 2  R
92adantr 261 . . 3  R  R  Er  X
10 simpr 103 . . 3  R  R
119, 10, 10ertr4d 6061 . 2  R  R
128, 11exlimddv 1775 1  R
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   class class class wbr 3755   dom cdm 4288    Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042
This theorem is referenced by:  iserd  6068  erth  6086  iinerm  6114  erinxp  6116
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