Theorem elqsg 6582

Description: Closed form of elqs 6583. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Assertion

Ref	Expression
elqsg	|- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )

Distinct variable groups:   x, A   x, B   x, R

Allowed substitution hint:   V( x)

Proof of Theorem elqsg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2184	. . 3 |- ( y = B -> ( y = [ x ] R <-> B = [ x ] R ) )
2	1	rexbidv 2478	. 2 |- ( y = B -> ( E. x e. A y = [ x ] R <-> E. x e. A B = [ x ] R ) )
3		df-qs 6538	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
4	2, 3	elab2g 2884	1 |- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   [cec 6530   /.cqs 6531

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-qs 6538

This theorem is referenced by:  elqs  6583  elqsi  6584  ecelqsg  6585