Theorem erexb 6645

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 4942	. . 3 |- ( R e. _V -> dom R e. _V )
2		erdm 6630	. . . 4 |- ( R Er A -> dom R = A )
3	2	eleq1d 2274	. . 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 6644	. 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 2176   _Vcvv 2772   dom cdm 4675   Er wer 6617

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-dm 4685  df-rn 4686  df-er 6620

This theorem is referenced by: (None)