Theorem elqsg 6639

Description: Closed form of elqs 6640. (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 2200	. . 3 |- ( y = B -> ( y = [ x ] R <-> B = [ x ] R ) )
2	1	rexbidv 2495	. 2 |- ( y = B -> ( E. x e. A y = [ x ] R <-> E. x e. A B = [ x ] R ) )
3		df-qs 6593	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
4	2, 3	elab2g 2907	1 |- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   E.wrex 2473   [cec 6585   /.cqs 6586

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-qs 6593

This theorem is referenced by:  elqs  6640  elqsi  6641  ecelqsg  6642  quselbasg  13300