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Theorem funfv 5947
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 5889 . . . . 5  |-  ( F `
 A )  e. 
_V
21unisn 4205 . . . 4  |-  U. {
( F `  A
) }  =  ( F `  A )
3 eqid 2471 . . . . . . 7  |-  dom  F  =  dom  F
4 df-fn 5592 . . . . . . 7  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
53, 4mpbiran2 933 . . . . . 6  |-  ( F  Fn  dom  F  <->  Fun  F )
6 fnsnfv 5940 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
75, 6sylanbr 481 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
87unieqd 4200 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  U. ( F " { A } ) )
92, 8syl5eqr 2519 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
109ex 441 . 2  |-  ( Fun 
F  ->  ( A  e.  dom  F  ->  ( F `  A )  =  U. ( F " { A } ) ) )
11 ndmfv 5903 . . 3  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
12 ndmima 5211 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F " { A } )  =  (/) )
1312unieqd 4200 . . . 4  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  = 
U. (/) )
14 uni0 4217 . . . 4  |-  U. (/)  =  (/)
1513, 14syl6eq 2521 . . 3  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  =  (/) )
1611, 15eqtr4d 2508 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  U. ( F " { A }
) )
1710, 16pm2.61d1 164 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   (/)c0 3722   {csn 3959   U.cuni 4190   dom cdm 4839   "cima 4842   Fun wfun 5583    Fn wfn 5584   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597
This theorem is referenced by:  funfv2  5948  fvun  5950  dffv2  5953
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