Theorem mrcfval 16268

Description: Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrcfval.f	|- F = (mrCls ` C )

Assertion

Ref	Expression
mrcfval	|- ( C e. (Moore ` X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )

Distinct variable groups:   x, F, s   x, C, s   x, X, s

Proof of Theorem mrcfval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrcfval.f	. 2 |- F = (mrCls ` C )
2		fvssunirn 6217	. . . . 5 |- (Moore ` X ) C_ U. ran Moore
3	2	sseli 3599	. . . 4 |- ( C e. (Moore ` X ) -> C e. U. ran Moore )
4		unieq 4444	. . . . . . 7 |- ( c = C -> U. c = U. C )
5	4	pweqd 4163	. . . . . 6 |- ( c = C -> ~P U. c = ~P U. C )
6		rabeq 3192	. . . . . . 7 |- ( c = C -> { s e. c | x C_ s } = { s e. C | x C_ s } )
7	6	inteqd 4480	. . . . . 6 |- ( c = C -> |^| { s e. c | x C_ s } = |^| { s e. C | x C_ s } )
8	5, 7	mpteq12dv 4733	. . . . 5 |- ( c = C -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) = ( x e. ~P U. C |-> |^| { s e. C | x C_ s } ) )
9		df-mrc 16247	. . . . 5 |- mrCls = ( c e. U. ran Moore |-> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) )
10		mreunirn 16261	. . . . . . . 8 |- ( c e. U. ran Moore <-> c e. (Moore ` U. c ) )
11		mrcflem 16266	. . . . . . . 8 |- ( c e. (Moore ` U. c ) -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) : ~P U. c --> c )
12	10, 11	sylbi 207	. . . . . . 7 |- ( c e. U. ran Moore -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) : ~P U. c --> c )
13		fssxp 6060	. . . . . . 7 |- ( ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) : ~P U. c --> c -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) C_ ( ~P U. c X. c ) )
14	12, 13	syl 17	. . . . . 6 |- ( c e. U. ran Moore -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) C_ ( ~P U. c X. c ) )
15		vuniex 6954	. . . . . . . 8 |- U. c e. _V
16	15	pwex 4848	. . . . . . 7 |- ~P U. c e. _V
17		vex 3203	. . . . . . 7 |- c e. _V
18	16, 17	xpex 6962	. . . . . 6 |- ( ~P U. c X. c ) e. _V
19		ssexg 4804	. . . . . 6 |- ( ( ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) C_ ( ~P U. c X. c ) /\ ( ~P U. c X. c ) e. _V ) -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) e. _V )
20	14, 18, 19	sylancl 694	. . . . 5 |- ( c e. U. ran Moore -> ( x e. ~P U. c |-> |^| { s e. c | x C_ s } ) e. _V )
21	8, 9, 20	fvmpt3 6286	. . . 4 |- ( C e. U. ran Moore -> (mrCls ` C ) = ( x e. ~P U. C |-> |^| { s e. C | x C_ s } ) )
22	3, 21	syl 17	. . 3 |- ( C e. (Moore ` X ) -> (mrCls ` C ) = ( x e. ~P U. C |-> |^| { s e. C | x C_ s } ) )
23		mreuni 16260	. . . . 5 |- ( C e. (Moore ` X ) -> U. C = X )
24	23	pweqd 4163	. . . 4 |- ( C e. (Moore ` X ) -> ~P U. C = ~P X )
25	24	mpteq1d 4738	. . 3 |- ( C e. (Moore ` X ) -> ( x e. ~P U. C |-> |^| { s e. C | x C_ s } ) = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )
26	22, 25	eqtrd 2656	. 2 |- ( C e. (Moore ` X ) -> (mrCls ` C ) = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )
27	1, 26	syl5eq 2668	1 |- ( C e. (Moore ` X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243

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  ax-un 6949

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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  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-f 5892  df-fv 5896  df-mre 16246  df-mrc 16247

This theorem is referenced by:  mrcf  16269  mrcval  16270  acsficl2d  17176  mrclsp  18989  mrccls  20883