Theorem elqsg 6479

Description: Closed form of elqs 6480. (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 2146	. . 3 |- ( y = B -> ( y = [ x ] R <-> B = [ x ] R ) )
2	1	rexbidv 2438	. 2 |- ( y = B -> ( E. x e. A y = [ x ] R <-> E. x e. A B = [ x ] R ) )
3		df-qs 6435	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
4	2, 3	elab2g 2831	1 |- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   [cec 6427   /.cqs 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-qs 6435

This theorem is referenced by:  elqs  6480  elqsi  6481  ecelqsg  6482