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Theorem elabrex 5779
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 1368 . . . 4  |- T.
2 csbeq1a 3081 . . . . . . 7  |-  ( x  =  z  ->  B  =  [_ z  /  x ]_ B )
32equcoms 1719 . . . . . 6  |-  ( z  =  x  ->  B  =  [_ z  /  x ]_ B )
4 trud 1380 . . . . . 6  |-  ( z  =  x  -> T.  )
53, 42thd 175 . . . . 5  |-  ( z  =  x  ->  ( B  =  [_ z  /  x ]_ B  <-> T.  )
)
65rspcev 2856 . . . 4  |-  ( ( x  e.  A  /\ T.  )  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
71, 6mpan2 425 . . 3  |-  ( x  e.  A  ->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
8 elabrex.1 . . . 4  |-  B  e. 
_V
9 eqeq1 2196 . . . . 5  |-  ( y  =  B  ->  (
y  =  [_ z  /  x ]_ B  <->  B  =  [_ z  /  x ]_ B ) )
109rexbidv 2491 . . . 4  |-  ( y  =  B  ->  ( E. z  e.  A  y  =  [_ z  /  x ]_ B  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B ) )
118, 10elab 2896 . . 3  |-  ( B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }  <->  E. z  e.  A  B  =  [_ z  /  x ]_ B )
127, 11sylibr 134 . 2  |-  ( x  e.  A  ->  B  e.  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B } )
13 nfv 1539 . . . 4  |-  F/ z  y  =  B
14 nfcsb1v 3105 . . . . 5  |-  F/_ x [_ z  /  x ]_ B
1514nfeq2 2344 . . . 4  |-  F/ x  y  =  [_ z  /  x ]_ B
162eqeq2d 2201 . . . 4  |-  ( x  =  z  ->  (
y  =  B  <->  y  =  [_ z  /  x ]_ B ) )
1713, 15, 16cbvrex 2715 . . 3  |-  ( E. x  e.  A  y  =  B  <->  E. z  e.  A  y  =  [_ z  /  x ]_ B )
1817abbii 2305 . 2  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. z  e.  A  y  =  [_ z  /  x ]_ B }
1912, 18eleqtrrdi 2283 1  |-  ( x  e.  A  ->  B  e.  { y  |  E. x  e.  A  y  =  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364   T. wtru 1365    e. wcel 2160   {cab 2175   E.wrex 2469   _Vcvv 2752   [_csb 3072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073
This theorem is referenced by:  eusvobj2  5883
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