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Theorem elfv 5423
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 5420 . . 3  |-  ( F `
 B )  = 
U. { x  | 
A. y ( B F y  <->  y  =  x ) }
21eleq2i 2207 . 2  |-  ( A  e.  ( F `  B )  <->  A  e.  U. { x  |  A. y ( B F y  <->  y  =  x ) } )
3 eluniab 3752 . 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 183 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:    /\ wa 103    <-> wb 104   A.wal 1330   E.wex 1469    e. wcel 1481   {cab 2126   U.cuni 3740   class class class wbr 3933   ` cfv 5127
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2689  df-sn 3534  df-uni 3741  df-iota 5092  df-fv 5135
This theorem is referenced by:  fv3  5448  relelfvdm  5457
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