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Theorem fneu 5348
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 5271 . . . 4  |-  ( Fun 
F  ->  E* y  B F y )
21adantr 451 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E* y  B F
y )
3 eldmg 4874 . . . . . 6  |-  ( B  e.  dom  F  -> 
( B  e.  dom  F  <->  E. y  B F
y ) )
43ibi 232 . . . . 5  |-  ( B  e.  dom  F  ->  E. y  B F
y )
54adantl 452 . . . 4  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E. y  B F
y )
6 exmoeu2 2186 . . . 4  |-  ( E. y  B F y  ->  ( E* y  B F y  <->  E! y  B F y ) )
75, 6syl 15 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( E* y  B F y  <->  E! y  B F y ) )
82, 7mpbid 201 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E! y  B F
y )
98funfni 5344 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   E!weu 2143   E*wmo 2144   class class class wbr 4023   dom cdm 4689   Fun wfun 5249    Fn wfn 5250
This theorem is referenced by:  fneu2  5349  fnbrfvb  5563  mapsn  6809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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