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Theorem erex 6578
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 6577 . . 3  |-  ( R  Er  A  ->  R  C_  ( A  X.  A
) )
2 xpexg 4755 . . . 4  |-  ( ( A  e.  V  /\  A  e.  V )  ->  ( A  X.  A
)  e.  _V )
32anidms 397 . . 3  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
4 ssexg 4157 . . 3  |-  ( ( R  C_  ( A  X.  A )  /\  ( A  X.  A )  e. 
_V )  ->  R  e.  _V )
51, 3, 4syl2an 289 . 2  |-  ( ( R  Er  A  /\  A  e.  V )  ->  R  e.  _V )
65ex 115 1  |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2160   _Vcvv 2752    C_ wss 3144    X. cxp 4639    Er wer 6551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-dm 4651  df-rn 4652  df-er 6554
This theorem is referenced by:  erexb  6579  qliftlem  6634  qusaddvallemg  12802  qusaddflemg  12803  qusaddval  12804  qusaddf  12805  qusmulval  12806  qusmulf  12807  qusgrp2  13048  eqgen  13159  qusrng  13305  qusring2  13409
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