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Theorem erex 7372
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 7371 . . 3  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
2 sqxpexg 6587 . . 3  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
3 ssexg 4540 . . 3  |-  ( ( R  C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  R  e.  _V )
41, 2, 3syl2an 475 . 2  |-  ( ( R  Er  A  /\  A  e.  V )  ->  R  e.  _V )
54ex 432 1  |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   _Vcvv 3059    C_ wss 3414    X. cxp 4821    Er wer 7345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-dm 4833  df-rn 4834  df-er 7348
This theorem is referenced by:  erexb  7373  qliftlem  7429  qshash  13790  qusaddvallem  15165  qusaddflem  15166  qusaddval  15167  qusaddf  15168  qusmulval  15169  qusmulf  15170  qusgrp2  16512  efgrelexlemb  17092  efgcpbllemb  17097  frgpuplem  17114  qusring2  17589  vitalilem2  22310  vitalilem3  22311
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