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Theorem erex 7121
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 7120 . . 3  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
2 xpexg 6506 . . . 4  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
32anidms 640 . . 3  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
4 ssexg 4435 . . 3  |-  ( ( R  C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  R  e.  _V )
51, 3, 4syl2an 474 . 2  |-  ( ( R  Er  A  /\  A  e.  V )  ->  R  e.  _V )
65ex 434 1  |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1761   _Vcvv 2970    C_ wss 3325    X. cxp 4834    Er wer 7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846  df-rn 4847  df-er 7097
This theorem is referenced by:  erexb  7122  qliftlem  7177  qshash  13286  divsaddvallem  14485  divsaddflem  14486  divsaddval  14487  divsaddf  14488  divsmulval  14489  divsmulf  14490  divsgrp2  15666  efgrelexlemb  16240  efgcpbllemb  16245  frgpuplem  16262  divsrng2  16702  vitalilem2  21048  vitalilem3  21049
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