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Theorem fneu 5691
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 5610 . . . 4  |-  ( Fun 
F  ->  E* y  B F y )
21adantr 465 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E* y  B F
y )
3 eldmg 5208 . . . . . 6  |-  ( B  e.  dom  F  -> 
( B  e.  dom  F  <->  E. y  B F
y ) )
43ibi 241 . . . . 5  |-  ( B  e.  dom  F  ->  E. y  B F
y )
54adantl 466 . . . 4  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E. y  B F
y )
6 exmoeu2 2312 . . . 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 210 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E! y  B F
y )
98funfni 5687 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1613    e. wcel 1819   E!weu 2283   E*wmo 2284   class class class wbr 4456   dom cdm 5008   Fun wfun 5588    Fn wfn 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-fun 5596  df-fn 5597
This theorem is referenced by:  fneu2  5692  fnbrfvb  5913  mapsn  7479
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