Theorem funfvdm 5740

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 5687	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
2		unisng 3931	. . 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 2232	. . . . 5 |- dom F = dom F
5		df-fn 5355	. . . . 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 5736	. . . 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 3925	. 2 |- ( ( Fun F /\ A e. dom F ) -> U. { ( F ` A ) } = U. ( F " { A } ) )
10	3, 9	eqtr3d 2267	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. ( F " { A } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   {csn 3689   U.cuni 3914   dom cdm 4749   "cima 4752   Fun wfun 5346   Fn wfn 5347   ` cfv 5352

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by:  funfvdm2  5741  fvun1  5743