Theorem funfv 6265

Description: A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)

Assertion

Ref	Expression
funfv	|- ( Fun F -> ( F ` A ) = U. ( F " { A } ) )

Proof of Theorem funfv

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 |- ( F ` A ) e. _V
2	1	unisn 4451	. . . 4 |- U. { ( F ` A ) } = ( F ` A )
3		eqid 2622	. . . . . . 7 |- dom F = dom F
4		df-fn 5891	. . . . . . 7 |- ( F Fn dom F <-> ( Fun F /\ dom F = dom F ) )
5	3, 4	mpbiran2 954	. . . . . 6 |- ( F Fn dom F <-> Fun F )
6		fnsnfv 6258	. . . . . 6 |- ( ( F Fn dom F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
7	5, 6	sylanbr 490	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> { ( F ` A ) } = ( F " { A } ) )
8	7	unieqd 4446	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> U. { ( F ` A ) } = U. ( F " { A } ) )
9	2, 8	syl5eqr 2670	. . 3 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = U. ( F " { A } ) )
10	9	ex 450	. 2 |- ( Fun F -> ( A e. dom F -> ( F ` A ) = U. ( F " { A } ) ) )
11		ndmfv 6218	. . 3 |- ( -. A e. dom F -> ( F ` A ) = (/) )
12		ndmima 5502	. . . . 5 |- ( -. A e. dom F -> ( F " { A } ) = (/) )
13	12	unieqd 4446	. . . 4 |- ( -. A e. dom F -> U. ( F " { A } ) = U. (/) )
14		uni0 4465	. . . 4 |- U. (/) = (/)
15	13, 14	syl6eq 2672	. . 3 |- ( -. A e. dom F -> U. ( F " { A } ) = (/) )
16	11, 15	eqtr4d 2659	. 2 |- ( -. A e. dom F -> ( F ` A ) = U. ( F " { A } ) )
17	10, 16	pm2.61d1 171	1 |- ( Fun F -> ( F ` A ) = U. ( F " { A } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   {csn 4177   U.cuni 4436   dom cdm 5114   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  funfv2  6266  fvun  6268  dffv2  6271