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Theorem elfv 5727
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 5724 . . 3  |-  ( F `
 B )  = 
U. { x  | 
A. y ( B F y  <->  y  =  x ) }
21eleq2i 2501 . 2  |-  ( A  e.  ( F `  B )  <->  A  e.  U. { x  |  A. y ( B F y  <->  y  =  x ) } )
3 eluniab 4028 . 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 242 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 178    /\ wa 360   A.wal 1550   E.wex 1551    e. wcel 1726   {cab 2423   U.cuni 4016   class class class wbr 4213   ` cfv 5455
This theorem is referenced by:  fv3  5745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-v 2959  df-sn 3821  df-uni 4017  df-iota 5419  df-fv 5463
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