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Theorem funfv 5790
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 5742 . . . . 5  |-  ( F `
 A )  e. 
_V
21unisn 4031 . . . 4  |-  U. {
( F `  A
) }  =  ( F `  A )
3 eqid 2436 . . . . . . 7  |-  dom  F  =  dom  F
4 df-fn 5457 . . . . . . 7  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
53, 4mpbiran2 886 . . . . . 6  |-  ( F  Fn  dom  F  <->  Fun  F )
6 fnsnfv 5786 . . . . . 6  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
75, 6sylanbr 460 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
87unieqd 4026 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  U. ( F " { A } ) )
92, 8syl5eqr 2482 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
109ex 424 . 2  |-  ( Fun 
F  ->  ( A  e.  dom  F  ->  ( F `  A )  =  U. ( F " { A } ) ) )
11 ndmfv 5755 . . 3  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
12 ndmima 5241 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F " { A } )  =  (/) )
1312unieqd 4026 . . . 4  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  = 
U. (/) )
14 uni0 4042 . . . 4  |-  U. (/)  =  (/)
1513, 14syl6eq 2484 . . 3  |-  ( -.  A  e.  dom  F  ->  U. ( F " { A } )  =  (/) )
1611, 15eqtr4d 2471 . 2  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  U. ( F " { A }
) )
1710, 16pm2.61d1 153 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   (/)c0 3628   {csn 3814   U.cuni 4015   dom cdm 4878   "cima 4881   Fun wfun 5448    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  funfv2  5791  fvun  5793  dffv2  5796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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