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Theorem funfv 5934
Description: A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
funfv  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )

Proof of Theorem funfv
StepHypRef Expression
1 fvex 5876 . . . . 5  |-  ( F `
 A )  e. 
_V
21unisn 4260 . . . 4  |-  U. {
( F `  A
) }  =  ( F `  A )
3 eqid 2467 . . . . . . 7  |-  dom  F  =  dom  F
4 df-fn 5591 . . . . . . 7  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
53, 4mpbiran2 917 . . . . . 6  |-  ( F  Fn  dom  F  <->  Fun  F )
6 fnsnfv 5927 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
75, 6sylanbr 473 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
87unieqd 4255 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  U. ( F " { A } ) )
92, 8syl5eqr 2522 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
109ex 434 . 2  |-  ( Fun 
F  ->  ( A  e.  dom  F  ->  ( F `  A )  =  U. ( F " { A } ) ) )
11 ndmfv 5890 . . 3  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
12 ndmima 5373 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F " { A } )  =  (/) )
1312unieqd 4255 . . . 4  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  = 
U. (/) )
14 uni0 4272 . . . 4  |-  U. (/)  =  (/)
1513, 14syl6eq 2524 . . 3  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  =  (/) )
1611, 15eqtr4d 2511 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  U. ( F " { A }
) )
1710, 16pm2.61d1 159 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   (/)c0 3785   {csn 4027   U.cuni 4245   dom cdm 4999   "cima 5002   Fun wfun 5582    Fn wfn 5583   ` cfv 5588
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-8 1769  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-pow 4625  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-sbc 3332  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-uni 4246  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-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596
This theorem is referenced by:  funfv2  5935  fvun  5937  dffv2  5940
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