Theorem funfvdm 5621

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

Step	Hyp	Ref	Expression
1		funfvex 5572	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
2		unisng 3853	. . 3 |- ( ( F ` A ) e. _V -> U. { ( F ` A ) } = ( F ` A ) )
3	1, 2	syl 14	. 2 |- ( ( Fun F /\ A e. dom F ) -> U. { ( F ` A ) } = ( F ` A ) )
4		eqid 2193	. . . . 5 |- dom F = dom F
5		df-fn 5258	. . . . 5 |- ( F Fn dom F <-> ( Fun F /\ dom F = dom F ) )
6	4, 5	mpbiran2 943	. . . 4 |- ( F Fn dom F <-> Fun F )
7		fnsnfv 5617	. . . 4 |- ( ( F Fn dom F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
8	6, 7	sylanbr 285	. . 3 |- ( ( Fun F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
9	8	unieqd 3847	. 2 |- ( ( Fun F /\ A e. dom F ) -> U. { ( F ` A ) } = U. ( F " { A } ) )
10	3, 9	eqtr3d 2228	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. ( F " { A } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   {csn 3619   U.cuni 3836   dom cdm 4660   "cima 4663   Fun wfun 5249   Fn wfn 5250   ` cfv 5255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by:  funfvdm2  5622  fvun1  5624