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Theorem erexb 6922
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 5122 . . 3  |-  ( R  e.  _V  ->  dom  R  e.  _V )
2 erdm 6907 . . . 4  |-  ( R  Er  A  ->  dom  R  =  A )
32eleq1d 2501 . . 3  |-  ( R  Er  A  ->  ( dom  R  e.  _V  <->  A  e.  _V ) )
41, 3syl5ib 211 . 2  |-  ( R  Er  A  ->  ( R  e.  _V  ->  A  e.  _V ) )
5 erex 6921 . 2  |-  ( R  Er  A  ->  ( A  e.  _V  ->  R  e.  _V ) )
64, 5impbid 184 1  |-  ( R  Er  A  ->  ( R  e.  _V  <->  A  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1725   _Vcvv 2948   dom cdm 4870    Er wer 6894
This theorem is referenced by:  prtex  26720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-er 6897
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