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Theorem elabrex 5781
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 1312 . . . 4  |-  T.
2 csbeq1a 3102 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
32eqcoms 2299 . . . . . 6  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
4 a1tru 1321 . . . . . 6  |-  ( z  =  x  ->  T.  )
53, 42thd 231 . . . . 5  |-  ( z  =  x  ->  ( B  =  [_ z  /  x ]_ B  <->  T.  )
)
65rspcev 2897 . . . 4  |-  ( ( x  e.  A  /\  T.  )  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
71, 6mpan2 652 . . 3  |-  ( x  e.  A  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
8 elabrex.1 . . . 4  |-  B  e. 
_V
9 eqeq1 2302 . . . . 5  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
109rexbidv 2577 . . . 4  |-  ( y  =  B  ->  ( E. z  e.  A  y  =  [_ z  /  x ]_ B  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B ) )
118, 10elab 2927 . . 3  |-  ( B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
127, 11sylibr 203 . 2  |-  ( x  e.  A  ->  B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B } )
13 nfv 1609 . . . 4  |-  F/ z  y  =  B
14 nfcsb1v 3126 . . . . 5  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2443 . . . 4  |-  F/ x  y  =  [_ z  /  x ]_ B
162eqeq2d 2307 . . . 4  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
1713, 15, 16cbvrex 2774 . . 3  |-  ( E. x  e.  A  y  =  B  <->  E. z  e.  A  y  =  [_ z  /  x ]_ B )
1817abbii 2408 . 2  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }
1912, 18syl6eleqr 2387 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 1307    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   [_csb 3094
This theorem is referenced by:  eusvobj2  6353  lss1d  15736  prdsxmetlem  17948  prdsbl  18053  itg2monolem1  19121  heibor1  26637  dihglblem5  32110
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-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-csb 3095
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