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Theorem fneu 5489
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 5410 . . . 4  |-  ( Fun 
F  ->  E* y  B F y )
21adantr 452 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E* y  B F
y )
3 eldmg 5005 . . . . . 6  |-  ( B  e.  dom  F  -> 
( B  e.  dom  F  <->  E. y  B F
y ) )
43ibi 233 . . . . 5  |-  ( B  e.  dom  F  ->  E. y  B F
y )
54adantl 453 . . . 4  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E. y  B F
y )
6 exmoeu2 2281 . . . 4  |-  ( E. y  B F y  ->  ( E* y  B F y  <->  E! y  B F y ) )
75, 6syl 16 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( E* y  B F y  <->  E! y  B F y ) )
82, 7mpbid 202 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E! y  B F
y )
98funfni 5485 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    e. wcel 1717   E!weu 2238   E*wmo 2239   class class class wbr 4153   dom cdm 4818   Fun wfun 5388    Fn wfn 5389
This theorem is referenced by:  fneu2  5490  fnbrfvb  5706  mapsn  6991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-fun 5396  df-fn 5397
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