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Theorem elabrex 5955
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
Hypothesis
Ref Expression
elabrex.1  |-  B  e. 
_V
Assertion
Ref Expression
elabrex  |-  ( x  e.  A  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
Distinct variable groups:    y, B    x, y, A
Allowed substitution hint:    B( x)

Proof of Theorem elabrex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tru 1373 . . . 4  |- T.
2 csbeq1a 3292 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
32equcoms 1733 . . . . . 6  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
4 a1tru 1385 . . . . . 6  |-  ( z  =  x  -> T.  )
53, 42thd 240 . . . . 5  |-  ( z  =  x  ->  ( B  =  [_ z  /  x ]_ B  <-> T.  )
)
65rspcev 3068 . . . 4  |-  ( ( x  e.  A  /\ T.  )  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
71, 6mpan2 671 . . 3  |-  ( x  e.  A  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
8 elabrex.1 . . . 4  |-  B  e. 
_V
9 eqeq1 2444 . . . . 5  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
109rexbidv 2731 . . . 4  |-  ( y  =  B  ->  ( E. z  e.  A  y  =  [_ z  /  x ]_ B  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B ) )
118, 10elab 3101 . . 3  |-  ( B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
127, 11sylibr 212 . 2  |-  ( x  e.  A  ->  B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B } )
13 nfv 1673 . . . 4  |-  F/ z  y  =  B
14 nfcsb1v 3299 . . . . 5  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2585 . . . 4  |-  F/ x  y  =  [_ z  /  x ]_ B
162eqeq2d 2449 . . . 4  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
1713, 15, 16cbvrex 2939 . . 3  |-  ( E. x  e.  A  y  =  B  <->  E. z  e.  A  y  =  [_ z  /  x ]_ B )
1817abbii 2550 . 2  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }
1912, 18syl6eleqr 2529 1  |-  ( x  e.  A  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   T. wtru 1370    e. wcel 1756   {cab 2424   E.wrex 2711   _Vcvv 2967   [_csb 3283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-v 2969  df-sbc 3182  df-csb 3284
This theorem is referenced by:  eusvobj2  6079  lss1d  17021  prdsxmetlem  19918  prdsbl  20041  itg2monolem1  21203  heibor1  28662  dihglblem5  34783
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