Theorem erex 6821

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

Step	Hyp	Ref	Expression
1		erssxp 6820	. . 3 |- ( R Er A -> R C_ ( A X. A ) )
2		xpexg 4884	. . . 4 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
3	2	anidms 401	. . 3 |- ( A e. V -> ( A X. A ) e. _V )
4		ssexg 4267	. . 3 |- ( ( R C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> R e. _V )
5	1, 3, 4	syl2an 289	. 2 |- ( ( R Er A /\ A e. V ) -> R e. _V )
6	5	ex 115	1 |- ( R Er A -> ( A e. V -> R e. _V ) )

Syntax hints:   -> wi 4   e. wcel 2209   _Vcvv 2821   C_ wss 3220   X. cxp 4767   Er wer 6794

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-er 6797

This theorem is referenced by:  erexb  6822  qliftlem  6877  qusaddvallemg  13631  qusaddflemg  13632  qusaddval  13633  qusaddf  13634  qusmulval  13635  qusmulf  13636  qusgrp2  13893  eqgen  14007  qusrng  14232  qusring2  14344