Theorem isacs1i 16318

Description: A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
isacs1i	|- ( ( X e. V /\ F : ~P X --> ~P X ) -> { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } e. (ACS ` X ) )

Distinct variable groups:   F, s   X, s

Allowed substitution hint:   V( s)

Proof of Theorem isacs1i

Dummy variables a t f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . 4 |- { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } C_ ~P X
2	1	a1i 11	. . 3 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } C_ ~P X )
3		inss1 3833	. . . . . 6 |- ( X i^i |^| t ) C_ X
4		elpw2g 4827	. . . . . 6 |- ( X e. V -> ( ( X i^i |^| t ) e. ~P X <-> ( X i^i |^| t ) C_ X ) )
5	3, 4	mpbiri 248	. . . . 5 |- ( X e. V -> ( X i^i |^| t ) e. ~P X )
6	5	ad2antrr 762	. . . 4 |- ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) -> ( X i^i |^| t ) e. ~P X )
7		imassrn 5477	. . . . . . . . 9 |- ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ ran F
8		frn 6053	. . . . . . . . . 10 |- ( F : ~P X --> ~P X -> ran F C_ ~P X )
9	8	adantl 482	. . . . . . . . 9 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> ran F C_ ~P X )
10	7, 9	syl5ss 3614	. . . . . . . 8 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ ~P X )
11	10	unissd 4462	. . . . . . 7 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ U. ~P X )
12		unipw 4918	. . . . . . 7 |- U. ~P X = X
13	11, 12	syl6sseq 3651	. . . . . 6 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ X )
14	13	adantr 481	. . . . 5 |- ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) -> U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ X )
15		inss2 3834	. . . . . . . . . . . . . 14 |- ( X i^i |^| t ) C_ |^| t
16		intss1 4492	. . . . . . . . . . . . . 14 |- ( a e. t -> |^| t C_ a )
17	15, 16	syl5ss 3614	. . . . . . . . . . . . 13 |- ( a e. t -> ( X i^i |^| t ) C_ a )
18	17	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) /\ a e. t ) -> ( X i^i |^| t ) C_ a )
19		sspwb 4917	. . . . . . . . . . . 12 |- ( ( X i^i |^| t ) C_ a <-> ~P ( X i^i |^| t ) C_ ~P a )
20	18, 19	sylib 208	. . . . . . . . . . 11 |- ( ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) /\ a e. t ) -> ~P ( X i^i |^| t ) C_ ~P a )
21		ssrin 3838	. . . . . . . . . . 11 |- ( ~P ( X i^i |^| t ) C_ ~P a -> ( ~P ( X i^i |^| t ) i^i Fin ) C_ ( ~P a i^i Fin ) )
22	20, 21	syl 17	. . . . . . . . . 10 |- ( ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) /\ a e. t ) -> ( ~P ( X i^i |^| t ) i^i Fin ) C_ ( ~P a i^i Fin ) )
23		imass2 5501	. . . . . . . . . 10 |- ( ( ~P ( X i^i |^| t ) i^i Fin ) C_ ( ~P a i^i Fin ) -> ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ ( F " ( ~P a i^i Fin ) ) )
24	22, 23	syl 17	. . . . . . . . 9 |- ( ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) /\ a e. t ) -> ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ ( F " ( ~P a i^i Fin ) ) )
25	24	unissd 4462	. . . . . . . 8 |- ( ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) /\ a e. t ) -> U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ U. ( F " ( ~P a i^i Fin ) ) )
26		ssel2 3598	. . . . . . . . . 10 |- ( ( t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } /\ a e. t ) -> a e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } )
27		pweq 4161	. . . . . . . . . . . . . . . 16 |- ( s = a -> ~P s = ~P a )
28	27	ineq1d 3813	. . . . . . . . . . . . . . 15 |- ( s = a -> ( ~P s i^i Fin ) = ( ~P a i^i Fin ) )
29	28	imaeq2d 5466	. . . . . . . . . . . . . 14 |- ( s = a -> ( F " ( ~P s i^i Fin ) ) = ( F " ( ~P a i^i Fin ) ) )
30	29	unieqd 4446	. . . . . . . . . . . . 13 |- ( s = a -> U. ( F " ( ~P s i^i Fin ) ) = U. ( F " ( ~P a i^i Fin ) ) )
31		id 22	. . . . . . . . . . . . 13 |- ( s = a -> s = a )
32	30, 31	sseq12d 3634	. . . . . . . . . . . 12 |- ( s = a -> ( U. ( F " ( ~P s i^i Fin ) ) C_ s <-> U. ( F " ( ~P a i^i Fin ) ) C_ a ) )
33	32	elrab 3363	. . . . . . . . . . 11 |- ( a e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> ( a e. ~P X /\ U. ( F " ( ~P a i^i Fin ) ) C_ a ) )
34	33	simprbi 480	. . . . . . . . . 10 |- ( a e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } -> U. ( F " ( ~P a i^i Fin ) ) C_ a )
35	26, 34	syl 17	. . . . . . . . 9 |- ( ( t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } /\ a e. t ) -> U. ( F " ( ~P a i^i Fin ) ) C_ a )
36	35	adantll 750	. . . . . . . 8 |- ( ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) /\ a e. t ) -> U. ( F " ( ~P a i^i Fin ) ) C_ a )
37	25, 36	sstrd 3613	. . . . . . 7 |- ( ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) /\ a e. t ) -> U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ a )
38	37	ralrimiva 2966	. . . . . 6 |- ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) -> A. a e. t U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ a )
39		ssint 4493	. . . . . 6 |- ( U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ |^| t <-> A. a e. t U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ a )
40	38, 39	sylibr 224	. . . . 5 |- ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) -> U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ |^| t )
41	14, 40	ssind 3837	. . . 4 |- ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) -> U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ ( X i^i |^| t ) )
42		pweq 4161	. . . . . . . . 9 |- ( s = ( X i^i |^| t ) -> ~P s = ~P ( X i^i |^| t ) )
43	42	ineq1d 3813	. . . . . . . 8 |- ( s = ( X i^i |^| t ) -> ( ~P s i^i Fin ) = ( ~P ( X i^i |^| t ) i^i Fin ) )
44	43	imaeq2d 5466	. . . . . . 7 |- ( s = ( X i^i |^| t ) -> ( F " ( ~P s i^i Fin ) ) = ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) )
45	44	unieqd 4446	. . . . . 6 |- ( s = ( X i^i |^| t ) -> U. ( F " ( ~P s i^i Fin ) ) = U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) )
46		id 22	. . . . . 6 |- ( s = ( X i^i |^| t ) -> s = ( X i^i |^| t ) )
47	45, 46	sseq12d 3634	. . . . 5 |- ( s = ( X i^i |^| t ) -> ( U. ( F " ( ~P s i^i Fin ) ) C_ s <-> U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ ( X i^i |^| t ) ) )
48	47	elrab 3363	. . . 4 |- ( ( X i^i |^| t ) e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> ( ( X i^i |^| t ) e. ~P X /\ U. ( F " ( ~P ( X i^i |^| t ) i^i Fin ) ) C_ ( X i^i |^| t ) ) )
49	6, 41, 48	sylanbrc 698	. . 3 |- ( ( ( X e. V /\ F : ~P X --> ~P X ) /\ t C_ { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } ) -> ( X i^i |^| t ) e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } )
50	2, 49	ismred2 16263	. 2 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } e. (Moore ` X ) )
51		fssxp 6060	. . . 4 |- ( F : ~P X --> ~P X -> F C_ ( ~P X X. ~P X ) )
52		pwexg 4850	. . . . 5 |- ( X e. V -> ~P X e. _V )
53		xpexg 6960	. . . . 5 |- ( ( ~P X e. _V /\ ~P X e. _V ) -> ( ~P X X. ~P X ) e. _V )
54	52, 52, 53	syl2anc 693	. . . 4 |- ( X e. V -> ( ~P X X. ~P X ) e. _V )
55		ssexg 4804	. . . 4 |- ( ( F C_ ( ~P X X. ~P X ) /\ ( ~P X X. ~P X ) e. _V ) -> F e. _V )
56	51, 54, 55	syl2anr 495	. . 3 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> F e. _V )
57		simpr 477	. . . 4 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> F : ~P X --> ~P X )
58		pweq 4161	. . . . . . . . . 10 |- ( s = t -> ~P s = ~P t )
59	58	ineq1d 3813	. . . . . . . . 9 |- ( s = t -> ( ~P s i^i Fin ) = ( ~P t i^i Fin ) )
60	59	imaeq2d 5466	. . . . . . . 8 |- ( s = t -> ( F " ( ~P s i^i Fin ) ) = ( F " ( ~P t i^i Fin ) ) )
61	60	unieqd 4446	. . . . . . 7 |- ( s = t -> U. ( F " ( ~P s i^i Fin ) ) = U. ( F " ( ~P t i^i Fin ) ) )
62		id 22	. . . . . . 7 |- ( s = t -> s = t )
63	61, 62	sseq12d 3634	. . . . . 6 |- ( s = t -> ( U. ( F " ( ~P s i^i Fin ) ) C_ s <-> U. ( F " ( ~P t i^i Fin ) ) C_ t ) )
64	63	elrab3 3364	. . . . 5 |- ( t e. ~P X -> ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( F " ( ~P t i^i Fin ) ) C_ t ) )
65	64	rgen 2922	. . . 4 |- A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( F " ( ~P t i^i Fin ) ) C_ t )
66	57, 65	jctir 561	. . 3 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> ( F : ~P X --> ~P X /\ A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( F " ( ~P t i^i Fin ) ) C_ t ) ) )
67		feq1 6026	. . . . 5 |- ( f = F -> ( f : ~P X --> ~P X <-> F : ~P X --> ~P X ) )
68		imaeq1 5461	. . . . . . . . 9 |- ( f = F -> ( f " ( ~P t i^i Fin ) ) = ( F " ( ~P t i^i Fin ) ) )
69	68	unieqd 4446	. . . . . . . 8 |- ( f = F -> U. ( f " ( ~P t i^i Fin ) ) = U. ( F " ( ~P t i^i Fin ) ) )
70	69	sseq1d 3632	. . . . . . 7 |- ( f = F -> ( U. ( f " ( ~P t i^i Fin ) ) C_ t <-> U. ( F " ( ~P t i^i Fin ) ) C_ t ) )
71	70	bibi2d 332	. . . . . 6 |- ( f = F -> ( ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( f " ( ~P t i^i Fin ) ) C_ t ) <-> ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( F " ( ~P t i^i Fin ) ) C_ t ) ) )
72	71	ralbidv 2986	. . . . 5 |- ( f = F -> ( A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( f " ( ~P t i^i Fin ) ) C_ t ) <-> A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( F " ( ~P t i^i Fin ) ) C_ t ) ) )
73	67, 72	anbi12d 747	. . . 4 |- ( f = F -> ( ( f : ~P X --> ~P X /\ A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( f " ( ~P t i^i Fin ) ) C_ t ) ) <-> ( F : ~P X --> ~P X /\ A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( F " ( ~P t i^i Fin ) ) C_ t ) ) ) )
74	73	spcegv 3294	. . 3 |- ( F e. _V -> ( ( F : ~P X --> ~P X /\ A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( F " ( ~P t i^i Fin ) ) C_ t ) ) -> E. f ( f : ~P X --> ~P X /\ A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( f " ( ~P t i^i Fin ) ) C_ t ) ) ) )
75	56, 66, 74	sylc 65	. 2 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> E. f ( f : ~P X --> ~P X /\ A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( f " ( ~P t i^i Fin ) ) C_ t ) ) )
76		isacs 16312	. 2 |- ( { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } e. (ACS ` X ) <-> ( { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } e. (Moore ` X ) /\ E. f ( f : ~P X --> ~P X /\ A. t e. ~P X ( t e. { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } <-> U. ( f " ( ~P t i^i Fin ) ) C_ t ) ) ) )
77	50, 75, 76	sylanbrc 698	1 |- ( ( X e. V /\ F : ~P X --> ~P X ) -> { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } e. (ACS ` X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   X. cxp 5112   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888   Fincfn 7955  Moorecmre 16242  ACScacs 16245

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-acs 16249

This theorem is referenced by:  acsfn  16320