Theorem erex 6153

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 6152	. . 3 |- ( R Er A -> R C_ ( A X. A ) )
2		xpexg 4470	. . . 4 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
3	2	anidms 389	. . 3 |- ( A e. V -> ( A X. A ) e. _V )
4		ssexg 3917	. . 3 |- ( ( R C_ ( A X. A ) /\ ( A X. A ) e. _V ) -> R e. _V )
5	1, 3, 4	syl2an 283	. 2 |- ( ( R Er A /\ A e. V ) -> R e. _V )
6	5	ex 113	1 |- ( R Er A -> ( A e. V -> R e. _V ) )

Syntax hints:   -> wi 4   e. wcel 1433   _Vcvv 2601   C_ wss 2973   X. cxp 4361   Er wer 6126

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-er 6129

This theorem is referenced by:  erexb  6154  qliftlem  6207