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Theorem elqsg 6156
Description: Closed form of elqs 6157. (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 2046 . . 3  |-  ( y  =  B  ->  (
y  =  [ x ] R  <->  B  =  [
x ] R ) )
21rexbidv 2327 . 2  |-  ( y  =  B  ->  ( E. x  e.  A  y  =  [ x ] R  <->  E. x  e.  A  B  =  [ x ] R ) )
3 df-qs 6112 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
42, 3elab2g 2689 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 98    = wceq 1243    e. wcel 1393   E.wrex 2307   [cec 6104   /.cqs 6105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-qs 6112
This theorem is referenced by:  elqs  6157  elqsi  6158  ecelqsg  6159
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