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Theorem erex 6684
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 6683 . . 3  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
2 xpexg 4800 . . . 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 4160 . . 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 1684   _Vcvv 2788    C_ wss 3152    X. cxp 4687    Er wer 6657
This theorem is referenced by:  erexb  6685  qliftlem  6739  qshash  12285  divsaddvallem  13453  divsaddflem  13454  divsaddval  13455  divsaddf  13456  divsmulval  13457  divsmulf  13458  divsgrp2  14613  efgrelexlemb  15059  efgcpbllemb  15064  frgpuplem  15081  divsrng2  15403  vitalilem2  18964  vitalilem3  18965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-er 6660
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