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Theorem elqsg 6727
Description: Closed form of elqs 6728. (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.
StepHypRef Expression
1 eqeq1 2302 . . 3  |-  ( y  =  B  ->  (
y  =  [ x ] R  <->  B  =  [
x ] R ) )
21rexbidv 2577 . 2  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  [ x ] R  <->  E. x  e.  A  B  =  [ x ] R ) )
3 df-qs 6682 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
42, 3elab2g 2929 1  |-  ( B  e.  V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   E.wrex 2557   [cec 6674   /.cqs 6675
This theorem is referenced by:  elqs  6728  elqsi  6729  ecelqsg  6730  elpi1  18559  pdiveql  26271  prtlem11  26837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-qs 6682
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