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Theorem funfvdm 5484
Description: A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)
Assertion
Ref Expression
funfvdm  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )

Proof of Theorem funfvdm
StepHypRef Expression
1 funfvex 5438 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
2 unisng 3753 . . 3  |-  ( ( F `  A )  e.  _V  ->  U. {
( F `  A
) }  =  ( F `  A ) )
31, 2syl 14 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  ( F `  A ) )
4 eqid 2139 . . . . 5  |-  dom  F  =  dom  F
5 df-fn 5126 . . . . 5  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
64, 5mpbiran2 925 . . . 4  |-  ( F  Fn  dom  F  <->  Fun  F )
7 fnsnfv 5480 . . . 4  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
86, 7sylanbr 283 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
98unieqd 3747 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  U. ( F " { A } ) )
103, 9eqtr3d 2174 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686   {csn 3527   U.cuni 3736   dom cdm 4539   "cima 4542   Fun wfun 5117    Fn wfn 5118   ` cfv 5123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by:  funfvdm2  5485  fvun1  5487
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