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Theorem funeu 5616
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 5609 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 releldm 5078 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  A  e.  dom  F )
31, 2sylan 473 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  A  e.  dom  F )
4 eldmg 5041 . . . 4  |-  ( A  e.  dom  F  -> 
( A  e.  dom  F  <->  E. y  A F
y ) )
54ibi 244 . . 3  |-  ( A  e.  dom  F  ->  E. y  A F
y )
63, 5syl 17 . 2  |-  ( ( Fun  F  /\  A F B )  ->  E. y  A F y )
7 funmo 5608 . . . 4  |-  ( Fun 
F  ->  E* y  A F y )
87adantr 466 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  E* y  A F y )
9 df-mo 2268 . . 3  |-  ( E* y  A F y  <-> 
( E. y  A F y  ->  E! y  A F y ) )
108, 9sylib 199 . 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 370   E.wex 1659    e. wcel 1867   E!weu 2263   E*wmo 2264   class class class wbr 4417   dom cdm 4845   Rel wrel 4850   Fun wfun 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-fun 5594
This theorem is referenced by:  funeu2  5617  funbrfv  5910  frege124d  36025
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