Theorem erex 6549

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 6548	. . 3 |- ( R Er A -> R C_ ( A X. A ) )
2		xpexg 4734	. . . 4 |- ( ( A e. V /\ A e. V ) -> ( A X. A ) e. _V )
3	2	anidms 397	. . 3 |- ( A e. V -> ( A X. A ) e. _V )
4		ssexg 4137	. . 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 2146   _Vcvv 2735   C_ wss 3127   X. cxp 4618   Er wer 6522

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631  df-er 6525

This theorem is referenced by:  erexb  6550  qliftlem  6603