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Theorem funfv 5932
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 5875 . . . . 5  |-  ( F `
 A )  e. 
_V
21unisn 4213 . . . 4  |-  U. {
( F `  A
) }  =  ( F `  A )
3 eqid 2451 . . . . . . 7  |-  dom  F  =  dom  F
4 df-fn 5585 . . . . . . 7  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
53, 4mpbiran2 930 . . . . . 6  |-  ( F  Fn  dom  F  <->  Fun  F )
6 fnsnfv 5925 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
75, 6sylanbr 476 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
87unieqd 4208 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  U. ( F " { A } ) )
92, 8syl5eqr 2499 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
109ex 436 . 2  |-  ( Fun 
F  ->  ( A  e.  dom  F  ->  ( F `  A )  =  U. ( F " { A } ) ) )
11 ndmfv 5889 . . 3  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
12 ndmima 5205 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F " { A } )  =  (/) )
1312unieqd 4208 . . . 4  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  = 
U. (/) )
14 uni0 4225 . . . 4  |-  U. (/)  =  (/)
1513, 14syl6eq 2501 . . 3  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  =  (/) )
1611, 15eqtr4d 2488 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  U. ( F " { A }
) )
1710, 16pm2.61d1 163 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   (/)c0 3731   {csn 3968   U.cuni 4198   dom cdm 4834   "cima 4837   Fun wfun 5576    Fn wfn 5577   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590
This theorem is referenced by:  funfv2  5933  fvun  5935  dffv2  5938
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