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Theorem fneu 5197
Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fneu  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
Distinct variable groups:    y, F    y, B
Allowed substitution hint:    A( y)

Proof of Theorem fneu
StepHypRef Expression
1 funmo 5108 . . . 4  |-  ( Fun 
F  ->  E* y  B F y )
21adantr 274 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E* y  B F
y )
3 eldmg 4704 . . . . . 6  |-  ( B  e.  dom  F  -> 
( B  e.  dom  F  <->  E. y  B F
y ) )
43ibi 175 . . . . 5  |-  ( B  e.  dom  F  ->  E. y  B F
y )
54adantl 275 . . . 4  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E. y  B F
y )
6 exmoeu2 2025 . . . 4  |-  ( E. y  B F y  ->  ( E* y  B F y  <->  E! y  B F y ) )
75, 6syl 14 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( E* y  B F y  <->  E! y  B F y ) )
82, 7mpbid 146 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E! y  B F
y )
98funfni 5193 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1453    e. wcel 1465   E!weu 1977   E*wmo 1978   class class class wbr 3899   dom cdm 4509   Fun wfun 5087    Fn wfn 5088
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-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-fun 5095  df-fn 5096
This theorem is referenced by:  fneu2  5198  fnbrfvb  5430  mapsn  6552
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