Theorem erexb 6726

Description: An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
erexb	|- ( R Er A -> ( R e. _V <-> A e. _V ) )

Proof of Theorem erexb

Step	Hyp	Ref	Expression
1		dmexg 4996	. . 3 |- ( R e. _V -> dom R e. _V )
2		erdm 6711	. . . 4 |- ( R Er A -> dom R = A )
3	2	eleq1d 2300	. . 3 |- ( R Er A -> ( dom R e. _V <-> A e. _V ) )
4	1, 3	imbitrid 154	. 2 |- ( R Er A -> ( R e. _V -> A e. _V ) )
5		erex 6725	. 2 |- ( R Er A -> ( A e. _V -> R e. _V ) )
6	4, 5	impbid 129	1 |- ( R Er A -> ( R e. _V <-> A e. _V ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2202   _Vcvv 2802   dom cdm 4725   Er wer 6698

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-er 6701

This theorem is referenced by: (None)