Theorem elqsg 6644

Description: Closed form of elqs 6645. (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 2203	. . 3 |- ( y = B -> ( y = [ x ] R <-> B = [ x ] R ) )
2	1	rexbidv 2498	. 2 |- ( y = B -> ( E. x e. A y = [ x ] R <-> E. x e. A B = [ x ] R ) )
3		df-qs 6598	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
4	2, 3	elab2g 2911	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 2167   E.wrex 2476   [cec 6590   /.cqs 6591

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-qs 6598

This theorem is referenced by:  elqs  6645  elqsi  6646  ecelqsg  6647  quselbasg  13360