Theorem funfvdm 5718

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 5665	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
2		unisng 3915	. . 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 2231	. . . . 5 |- dom F = dom F
5		df-fn 5336	. . . . 5 |- ( F Fn dom F <-> ( Fun F /\ dom F = dom F ) )
6	4, 5	mpbiran2 950	. . . 4 |- ( F Fn dom F <-> Fun F )
7		fnsnfv 5714	. . . 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 3909	. 2 |- ( ( Fun F /\ A e. dom F ) -> U. { ( F ` A ) } = U. ( F " { A } ) )
10	3, 9	eqtr3d 2266	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. ( F " { A } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   {csn 3673   U.cuni 3898   dom cdm 4731   "cima 4734   Fun wfun 5327   Fn wfn 5328   ` cfv 5333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  funfvdm2  5719  fvun1  5721