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Theorem elabrex 5627
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 1320 . . . 4  |- T.
2 csbeq1a 2983 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
32equcoms 1669 . . . . . 6  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
4 a1tru 1332 . . . . . 6  |-  ( z  =  x  -> T.  )
53, 42thd 174 . . . . 5  |-  ( z  =  x  ->  ( B  =  [_ z  /  x ]_ B  <-> T.  )
)
65rspcev 2763 . . . 4  |-  ( ( x  e.  A  /\ T.  )  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
71, 6mpan2 421 . . 3  |-  ( x  e.  A  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
8 elabrex.1 . . . 4  |-  B  e. 
_V
9 eqeq1 2124 . . . . 5  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
109rexbidv 2415 . . . 4  |-  ( y  =  B  ->  ( E. z  e.  A  y  =  [_ z  /  x ]_ B  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B ) )
118, 10elab 2802 . . 3  |-  ( B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
127, 11sylibr 133 . 2  |-  ( x  e.  A  ->  B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B } )
13 nfv 1493 . . . 4  |-  F/ z  y  =  B
14 nfcsb1v 3005 . . . . 5  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2270 . . . 4  |-  F/ x  y  =  [_ z  /  x ]_ B
162eqeq2d 2129 . . . 4  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
1713, 15, 16cbvrex 2628 . . 3  |-  ( E. x  e.  A  y  =  B  <->  E. z  e.  A  y  =  [_ z  /  x ]_ B )
1817abbii 2233 . 2  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }
1912, 18eleqtrrdi 2211 1  |-  ( x  e.  A  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   T. wtru 1317    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660   [_csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by:  eusvobj2  5728
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