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Theorem erex 6700
Description: An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erex  |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V ) )

Proof of Theorem erex
StepHypRef Expression
1 erssxp 6699 . . 3  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
2 xpexg 4816 . . . 4  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
32anidms 626 . . 3  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
4 ssexg 4176 . . 3  |-  ( ( R  C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  R  e.  _V )
51, 3, 4syl2an 463 . 2  |-  ( ( R  Er  A  /\  A  e.  V )  ->  R  e.  _V )
65ex 423 1  |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801    C_ wss 3165    X. cxp 4703    Er wer 6673
This theorem is referenced by:  erexb  6701  qliftlem  6755  qshash  12301  divsaddvallem  13469  divsaddflem  13470  divsaddval  13471  divsaddf  13472  divsmulval  13473  divsmulf  13474  divsgrp2  14629  efgrelexlemb  15075  efgcpbllemb  15080  frgpuplem  15097  divsrng2  15419  vitalilem2  18980  vitalilem3  18981
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
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