ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  erexb Unicode version

Theorem erexb 6457
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 4806 . . 3  |-  ( R  e.  _V  ->  dom  R  e.  _V )
2 erdm 6442 . . . 4  |-  ( R  Er  A  ->  dom  R  =  A )
32eleq1d 2208 . . 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 6456 . 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 1480   _Vcvv 2686   dom cdm 4542    Er wer 6429
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-xp 4548  df-rel 4549  df-cnv 4550  df-dm 4552  df-rn 4553  df-er 6432
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator