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Theorem funeu 5260
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 5252 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 4880 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
31, 2sylan 283 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  A  e.  dom  F )
4 eldmg 4840 . . . 4  |-  ( A  e.  dom  F  -> 
( A  e.  dom  F  <->  E. y  A F
y ) )
54ibi 176 . . 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 5250 . . . 4  |-  ( Fun 
F  ->  E* y  A F y )
87adantr 276 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  E* y  A F y )
9 df-mo 2042 . . 3  |-  ( E* y  A F y  <-> 
( E. y  A F y  ->  E! y  A F y ) )
108, 9sylib 122 . 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 104   E.wex 1503   E!weu 2038   E*wmo 2039    e. wcel 2160   class class class wbr 4018   dom cdm 4644   Rel wrel 4649   Fun wfun 5229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-fun 5237
This theorem is referenced by:  funeu2  5261  funbrfv  5575
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