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Theorem funeu 5612
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 5605 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 5235 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
31, 2sylan 471 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  A  e.  dom  F )
4 eldmg 5198 . . . 4  |-  ( A  e.  dom  F  -> 
( A  e.  dom  F  <->  E. y  A F
y ) )
54ibi 241 . . 3  |-  ( A  e.  dom  F  ->  E. y  A F
y )
63, 5syl 16 . 2  |-  ( ( Fun  F  /\  A F B )  ->  E. y  A F y )
7 funmo 5604 . . . 4  |-  ( Fun 
F  ->  E* y  A F y )
87adantr 465 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  E* y  A F y )
9 df-mo 2280 . . 3  |-  ( E* y  A F y  <-> 
( E. y  A F y  ->  E! y  A F y ) )
108, 9sylib 196 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( E. y  A F
y  ->  E! y  A F y ) )
116, 10mpd 15 1  |-  ( ( Fun  F  /\  A F B )  ->  E! y  A F y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1596    e. wcel 1767   E!weu 2275   E*wmo 2276   class class class wbr 4447   dom cdm 4999   Rel wrel 5004   Fun wfun 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-fun 5590
This theorem is referenced by:  funeu2  5613  funbrfv  5906
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