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Theorem erexb 6422
Description: An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erexb  |-  ( R  Er  A  ->  ( R  e.  _V  <->  A  e.  _V ) )

Proof of Theorem erexb
StepHypRef Expression
1 dmexg 4773 . . 3  |-  ( R  e.  _V  ->  dom  R  e.  _V )
2 erdm 6407 . . . 4  |-  ( R  Er  A  ->  dom  R  =  A )
32eleq1d 2186 . . 3  |-  ( R  Er  A  ->  ( dom  R  e.  _V  <->  A  e.  _V ) )
41, 3syl5ib 153 . 2  |-  ( R  Er  A  ->  ( R  e.  _V  ->  A  e.  _V ) )
5 erex 6421 . 2  |-  ( R  Er  A  ->  ( A  e.  _V  ->  R  e.  _V ) )
64, 5impbid 128 1  |-  ( R  Er  A  ->  ( R  e.  _V  <->  A  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1465   _Vcvv 2660   dom cdm 4509    Er wer 6394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-er 6397
This theorem is referenced by: (None)
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