MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  erexb Unicode version

Theorem erexb 6701
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 4955 . . 3  |-  ( R  e.  _V  ->  dom  R  e.  _V )
2 erdm 6686 . . . 4  |-  ( R  Er  A  ->  dom  R  =  A )
32eleq1d 2362 . . 3  |-  ( R  Er  A  ->  ( dom  R  e.  _V  <->  A  e.  _V ) )
41, 3syl5ib 210 . 2  |-  ( R  Er  A  ->  ( R  e.  _V  ->  A  e.  _V ) )
5 erex 6700 . 2  |-  ( R  Er  A  ->  ( A  e.  _V  ->  R  e.  _V ) )
64, 5impbid 183 1  |-  ( R  Er  A  ->  ( R  e.  _V  <->  A  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1696   _Vcvv 2801   dom cdm 4705    Er wer 6673
This theorem is referenced by:  prtex  26851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-er 6676
  Copyright terms: Public domain W3C validator