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Theorem erex 7128
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 7127 . . 3  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
2 xpexg 6510 . . . 4  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
32anidms 645 . . 3  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
4 ssexg 4441 . . 3  |-  ( ( R  C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  R  e.  _V )
51, 3, 4syl2an 477 . 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 1756   _Vcvv 2975    C_ wss 3331    X. cxp 4841    Er wer 7101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-xp 4849  df-rel 4850  df-cnv 4851  df-dm 4853  df-rn 4854  df-er 7104
This theorem is referenced by:  erexb  7129  qliftlem  7184  qshash  13293  divsaddvallem  14492  divsaddflem  14493  divsaddval  14494  divsaddf  14495  divsmulval  14496  divsmulf  14497  divsgrp2  15676  efgrelexlemb  16250  efgcpbllemb  16255  frgpuplem  16272  divsrng2  16715  vitalilem2  21092  vitalilem3  21093
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