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Theorem funfvdm 5268
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 5223 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  _V )
2 unisng 3626 . . 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 2082 . . . . 5  |-  dom  F  =  dom  F
5 df-fn 4935 . . . . 5  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
64, 5mpbiran2 883 . . . 4  |-  ( F  Fn  dom  F  <->  Fun  F )
7 fnsnfv 5264 . . . 4  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
86, 7sylanbr 279 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
98unieqd 3620 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. { ( F `  A ) }  =  U. ( F " { A } ) )
103, 9eqtr3d 2116 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   {csn 3406   U.cuni 3609   dom cdm 4371   "cima 4374   Fun wfun 4926    Fn wfn 4927   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940
This theorem is referenced by:  funfvdm2  5269  fvun1  5271
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