Theorem isacs 16312

Description: A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Assertion

Ref	Expression
isacs	|- ( C e. (ACS ` X ) <-> ( C e. (Moore ` X ) /\ E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )

Distinct variable groups:   C, f, s   f, X, s

Proof of Theorem isacs

Dummy variables c x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6221	. 2 |- ( C e. (ACS ` X ) -> X e. _V )
2		elfvex 6221	. . 3 |- ( C e. (Moore ` X ) -> X e. _V )
3	2	adantr 481	. 2 |- ( ( C e. (Moore ` X ) /\ E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) -> X e. _V )
4		fveq2 6191	. . . . . 6 |- ( x = X -> (Moore ` x ) = (Moore ` X ) )
5		pweq 4161	. . . . . . . . 9 |- ( x = X -> ~P x = ~P X )
6	5, 5	feq23d 6040	. . . . . . . 8 |- ( x = X -> ( f : ~P x --> ~P x <-> f : ~P X --> ~P X ) )
7	5	raleqdv 3144	. . . . . . . 8 |- ( x = X -> ( A. s e. ~P x ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) <-> A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) )
8	6, 7	anbi12d 747	. . . . . . 7 |- ( x = X -> ( ( f : ~P x --> ~P x /\ A. s e. ~P x ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) <-> ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )
9	8	exbidv 1850	. . . . . 6 |- ( x = X -> ( E. f ( f : ~P x --> ~P x /\ A. s e. ~P x ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) <-> E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )
10	4, 9	rabeqbidv 3195	. . . . 5 |- ( x = X -> { c e. (Moore ` x ) | E. f ( f : ~P x --> ~P x /\ A. s e. ~P x ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) } = { c e. (Moore ` X ) | E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) } )
11		df-acs 16249	. . . . 5 |- ACS = ( x e. _V |-> { c e. (Moore ` x ) | E. f ( f : ~P x --> ~P x /\ A. s e. ~P x ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) } )
12		fvex 6201	. . . . . 6 |- (Moore ` X ) e. _V
13	12	rabex 4813	. . . . 5 |- { c e. (Moore ` X ) | E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) } e. _V
14	10, 11, 13	fvmpt 6282	. . . 4 |- ( X e. _V -> (ACS ` X ) = { c e. (Moore ` X ) | E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) } )
15	14	eleq2d 2687	. . 3 |- ( X e. _V -> ( C e. (ACS ` X ) <-> C e. { c e. (Moore ` X ) | E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) } ) )
16		eleq2 2690	. . . . . . . 8 |- ( c = C -> ( s e. c <-> s e. C ) )
17	16	bibi1d 333	. . . . . . 7 |- ( c = C -> ( ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) <-> ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) )
18	17	ralbidv 2986	. . . . . 6 |- ( c = C -> ( A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) <-> A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) )
19	18	anbi2d 740	. . . . 5 |- ( c = C -> ( ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) <-> ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )
20	19	exbidv 1850	. . . 4 |- ( c = C -> ( E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) <-> E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )
21	20	elrab 3363	. . 3 |- ( C e. { c e. (Moore ` X ) | E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. c <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) } <-> ( C e. (Moore ` X ) /\ E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )
22	15, 21	syl6bb 276	. 2 |- ( X e. _V -> ( C e. (ACS ` X ) <-> ( C e. (Moore ` X ) /\ E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) ) )
23	1, 3, 22	pm5.21nii 368	1 |- ( C e. (ACS ` X ) <-> ( C e. (Moore ` X ) /\ E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )

Syntax hints:   <-> 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   "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

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-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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-acs 16249

This theorem is referenced by:  acsmre  16313  isacs2  16314  isacs1i  16318  mreacs  16319