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Theorem elabrex 5944
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 1327 . . . 4  |-  T.
2 csbeq1a 3219 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
32equcoms 1689 . . . . . 6  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
4 a1tru 1336 . . . . . 6  |-  ( z  =  x  ->  T.  )
53, 42thd 232 . . . . 5  |-  ( z  =  x  ->  ( B  =  [_ z  /  x ]_ B  <->  T.  )
)
65rspcev 3012 . . . 4  |-  ( ( x  e.  A  /\  T.  )  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
71, 6mpan2 653 . . 3  |-  ( x  e.  A  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
8 elabrex.1 . . . 4  |-  B  e. 
_V
9 eqeq1 2410 . . . . 5  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
109rexbidv 2687 . . . 4  |-  ( y  =  B  ->  ( E. z  e.  A  y  =  [_ z  /  x ]_ B  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B ) )
118, 10elab 3042 . . 3  |-  ( B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
127, 11sylibr 204 . 2  |-  ( x  e.  A  ->  B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B } )
13 nfv 1626 . . . 4  |-  F/ z  y  =  B
14 nfcsb1v 3243 . . . . 5  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2551 . . . 4  |-  F/ x  y  =  [_ z  /  x ]_ B
162eqeq2d 2415 . . . 4  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
1713, 15, 16cbvrex 2889 . . 3  |-  ( E. x  e.  A  y  =  B  <->  E. z  e.  A  y  =  [_ z  /  x ]_ B )
1817abbii 2516 . 2  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }
1912, 18syl6eleqr 2495 1  |-  ( x  e.  A  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    T. wtru 1322    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667   _Vcvv 2916   [_csb 3211
This theorem is referenced by:  eusvobj2  6541  lss1d  15994  prdsxmetlem  18351  prdsbl  18474  itg2monolem1  19595  heibor1  26409  dihglblem5  31781
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-ral 2671  df-rex 2672  df-v 2918  df-sbc 3122  df-csb 3212
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