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Theorem fneu 5302
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 5213 . . . 4  |-  ( Fun 
F  ->  E* y  B F y )
21adantr 274 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  E* y  B F
y )
3 eldmg 4806 . . . . . 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 2067 . . . 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 5298 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 1485   E!weu 2019   E*wmo 2020    e. wcel 2141   class class class wbr 3989   dom cdm 4611   Fun wfun 5192    Fn wfn 5193
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-fun 5200  df-fn 5201
This theorem is referenced by:  fneu2  5303  fnbrfvb  5537  mapsn  6668
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