Theorem acsfiel 16315

Description: A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Hypothesis

Ref	Expression
isacs2.f	|- F = (mrCls ` C )

Assertion

Ref	Expression
acsfiel	|- ( C e. (ACS ` X ) -> ( S e. C <-> ( S C_ X /\ A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) ) )

Distinct variable groups:   y, C   y, F   y, S   y, X

Proof of Theorem acsfiel

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		acsmre 16313	. . . . 5 |- ( C e. (ACS ` X ) -> C e. (Moore ` X ) )
2		mress 16253	. . . . 5 |- ( ( C e. (Moore ` X ) /\ S e. C ) -> S C_ X )
3	1, 2	sylan 488	. . . 4 |- ( ( C e. (ACS ` X ) /\ S e. C ) -> S C_ X )
4	3	ex 450	. . 3 |- ( C e. (ACS ` X ) -> ( S e. C -> S C_ X ) )
5	4	pm4.71rd 667	. 2 |- ( C e. (ACS ` X ) -> ( S e. C <-> ( S C_ X /\ S e. C ) ) )
6		elfvdm 6220	. . . . . 6 |- ( C e. (ACS ` X ) -> X e. dom ACS )
7		elpw2g 4827	. . . . . 6 |- ( X e. dom ACS -> ( S e. ~P X <-> S C_ X ) )
8	6, 7	syl 17	. . . . 5 |- ( C e. (ACS ` X ) -> ( S e. ~P X <-> S C_ X ) )
9	8	biimpar 502	. . . 4 |- ( ( C e. (ACS ` X ) /\ S C_ X ) -> S e. ~P X )
10		isacs2.f	. . . . . . 7 |- F = (mrCls ` C )
11	10	isacs2 16314	. . . . . 6 |- ( C e. (ACS ` X ) <-> ( C e. (Moore ` X ) /\ A. s e. ~P X ( s e. C <-> A. y e. ( ~P s i^i Fin ) ( F ` y ) C_ s ) ) )
12	11	simprbi 480	. . . . 5 |- ( C e. (ACS ` X ) -> A. s e. ~P X ( s e. C <-> A. y e. ( ~P s i^i Fin ) ( F ` y ) C_ s ) )
13	12	adantr 481	. . . 4 |- ( ( C e. (ACS ` X ) /\ S C_ X ) -> A. s e. ~P X ( s e. C <-> A. y e. ( ~P s i^i Fin ) ( F ` y ) C_ s ) )
14		eleq1 2689	. . . . . 6 |- ( s = S -> ( s e. C <-> S e. C ) )
15		pweq 4161	. . . . . . . 8 |- ( s = S -> ~P s = ~P S )
16	15	ineq1d 3813	. . . . . . 7 |- ( s = S -> ( ~P s i^i Fin ) = ( ~P S i^i Fin ) )
17		sseq2 3627	. . . . . . 7 |- ( s = S -> ( ( F ` y ) C_ s <-> ( F ` y ) C_ S ) )
18	16, 17	raleqbidv 3152	. . . . . 6 |- ( s = S -> ( A. y e. ( ~P s i^i Fin ) ( F ` y ) C_ s <-> A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) )
19	14, 18	bibi12d 335	. . . . 5 |- ( s = S -> ( ( s e. C <-> A. y e. ( ~P s i^i Fin ) ( F ` y ) C_ s ) <-> ( S e. C <-> A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) ) )
20	19	rspcva 3307	. . . 4 |- ( ( S e. ~P X /\ A. s e. ~P X ( s e. C <-> A. y e. ( ~P s i^i Fin ) ( F ` y ) C_ s ) ) -> ( S e. C <-> A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) )
21	9, 13, 20	syl2anc 693	. . 3 |- ( ( C e. (ACS ` X ) /\ S C_ X ) -> ( S e. C <-> A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) )
22	21	pm5.32da 673	. 2 |- ( C e. (ACS ` X ) -> ( ( S C_ X /\ S e. C ) <-> ( S C_ X /\ A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) ) )
23	5, 22	bitrd 268	1 |- ( C e. (ACS ` X ) -> ( S e. C <-> ( S C_ X /\ A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   dom cdm 5114   ` cfv 5888   Fincfn 7955  Moorecmre 16242  mrClscmrc 16243  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-csb 3534  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-iun 4522  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  df-acs 16249

This theorem is referenced by:  acsfiel2  16316  isacs3lem  17166