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Theorem elfv 5539
Description: Membership in a function value. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
elfv  |-  ( A  e.  ( F `  B )  <->  E. x
( A  e.  x  /\  A. y ( B F y  <->  y  =  x ) ) )
Distinct variable groups:    x, A    x, y, B    x, F, y
Allowed substitution hint:    A( y)

Proof of Theorem elfv
StepHypRef Expression
1 fv2 5536 . . 3  |-  ( F `
 B )  = 
U. { x  | 
A. y ( B F y  <->  y  =  x ) }
21eleq2i 2360 . 2  |-  ( A  e.  ( F `  B )  <->  A  e.  U. { x  |  A. y ( B F y  <->  y  =  x ) } )
3 eluniab 3855 . 2  |-  ( A  e.  U. { x  |  A. y ( B F y  <->  y  =  x ) }  <->  E. x
( A  e.  x  /\  A. y ( B F y  <->  y  =  x ) ) )
42, 3bitri 240 1  |-  ( A  e.  ( F `  B )  <->  E. x
( A  e.  x  /\  A. y ( B F y  <->  y  =  x ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   U.cuni 3843   class class class wbr 4039   ` cfv 5271
This theorem is referenced by:  fv3  5557
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-rex 2562  df-v 2803  df-sn 3659  df-uni 3844  df-iota 5235  df-fv 5279
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