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Theorem elqsg 6915
Description: Closed form of elqs 6916. (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 2410 . . 3  |-  ( y  =  B  ->  (
y  =  [ x ] R  <->  B  =  [
x ] R ) )
21rexbidv 2687 . 2  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  [ x ] R  <->  E. x  e.  A  B  =  [ x ] R ) )
3 df-qs 6870 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
42, 3elab2g 3044 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 177    = wceq 1649    e. wcel 1721   E.wrex 2667   [cec 6862   /.cqs 6863
This theorem is referenced by:  elqs  6916  elqsi  6917  ecelqsg  6918  elpi1  19023  prtlem11  26605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-v 2918  df-qs 6870
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