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Theorem funeu 5106
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
funeu  |-  ( ( Fun  F  /\  A F B )  ->  E! y  A F y )
Distinct variable groups:    y, A    y, F
Allowed substitution hint:    B( y)

Proof of Theorem funeu
StepHypRef Expression
1 funrel 5098 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 4734 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
31, 2sylan 279 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  A  e.  dom  F )
4 eldmg 4694 . . . 4  |-  ( A  e.  dom  F  -> 
( A  e.  dom  F  <->  E. y  A F
y ) )
54ibi 175 . . 3  |-  ( A  e.  dom  F  ->  E. y  A F
y )
63, 5syl 14 . 2  |-  ( ( Fun  F  /\  A F B )  ->  E. y  A F y )
7 funmo 5096 . . . 4  |-  ( Fun 
F  ->  E* y  A F y )
87adantr 272 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  E* y  A F y )
9 df-mo 1979 . . 3  |-  ( E* y  A F y  <-> 
( E. y  A F y  ->  E! y  A F y ) )
108, 9sylib 121 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( E. y  A F
y  ->  E! y  A F y ) )
116, 10mpd 13 1  |-  ( ( Fun  F  /\  A F B )  ->  E! y  A F y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1451    e. wcel 1463   E!weu 1975   E*wmo 1976   class class class wbr 3895   dom cdm 4499   Rel wrel 4504   Fun wfun 5075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-fun 5083
This theorem is referenced by:  funeu2  5107  funbrfv  5414
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