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Theorem funfv 5602
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 5555 . . . . 5  |-  ( F `
 A )  e. 
_V
21unisn 3859 . . . 4  |-  U. {
( F `  A
) }  =  ( F `  A )
3 eqid 2296 . . . . . . 7  |-  dom  F  =  dom  F
4 df-fn 5274 . . . . . . 7  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
53, 4mpbiran2 885 . . . . . 6  |-  ( F  Fn  dom  F  <->  Fun  F )
6 fnsnfv 5598 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
75, 6sylanbr 459 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
87unieqd 3854 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  U. ( F " { A } ) )
92, 8syl5eqr 2342 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
109ex 423 . 2  |-  ( Fun 
F  ->  ( A  e.  dom  F  ->  ( F `  A )  =  U. ( F " { A } ) ) )
11 ndmfv 5568 . . 3  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
12 ndmima 5066 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F " { A } )  =  (/) )
1312unieqd 3854 . . . 4  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  = 
U. (/) )
14 uni0 3870 . . . 4  |-  U. (/)  =  (/)
1513, 14syl6eq 2344 . . 3  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  =  (/) )
1611, 15eqtr4d 2331 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  U. ( F " { A }
) )
1710, 16pm2.61d1 151 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   (/)c0 3468   {csn 3653   U.cuni 3843   dom cdm 4705   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  funfv2  5603  fvun  5605  dffv2  5608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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