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Theorem funfvdm 5416
Description: A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
Assertion
Ref Expression
funfvdm  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )

Proof of Theorem funfvdm
StepHypRef Expression
1 funfvex 5370 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
2 unisng 3700 . . 3  |-  ( ( F `  A )  e.  _V  ->  U. {
( F `  A
) }  =  ( F `  A ) )
31, 2syl 14 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  ( F `  A ) )
4 eqid 2100 . . . . 5  |-  dom  F  =  dom  F
5 df-fn 5062 . . . . 5  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
64, 5mpbiran2 893 . . . 4  |-  ( F  Fn  dom  F  <->  Fun  F )
7 fnsnfv 5412 . . . 4  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
86, 7sylanbr 281 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
98unieqd 3694 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  U. ( F " { A } ) )
103, 9eqtr3d 2134 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299    e. wcel 1448   _Vcvv 2641   {csn 3474   U.cuni 3683   dom cdm 4477   "cima 4480   Fun wfun 5053    Fn wfn 5054   ` cfv 5059
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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067
This theorem is referenced by:  funfvdm2  5417  fvun1  5419
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