Theorem isnacs2 37269

Description: Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Hypothesis

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

Assertion

Ref	Expression
isnacs2	|- ( C e. (NoeACS ` X ) <-> ( C e. (ACS ` X ) /\ ( F " ( ~P X i^i Fin ) ) = C ) )

Proof of Theorem isnacs2

Dummy variables g s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnacs.f	. . 3 |- F = (mrCls ` C )
2	1	isnacs 37267	. 2 |- ( C e. (NoeACS ` X ) <-> ( C e. (ACS ` X ) /\ A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) )
3		acsmre 16313	. . . . . . . . 9 |- ( C e. (ACS ` X ) -> C e. (Moore ` X ) )
4	1	mrcf 16269	. . . . . . . . 9 |- ( C e. (Moore ` X ) -> F : ~P X --> C )
5		ffn 6045	. . . . . . . . 9 |- ( F : ~P X --> C -> F Fn ~P X )
6	3, 4, 5	3syl 18	. . . . . . . 8 |- ( C e. (ACS ` X ) -> F Fn ~P X )
7		inss1 3833	. . . . . . . 8 |- ( ~P X i^i Fin ) C_ ~P X
8		fvelimab 6253	. . . . . . . 8 |- ( ( F Fn ~P X /\ ( ~P X i^i Fin ) C_ ~P X ) -> ( s e. ( F " ( ~P X i^i Fin ) ) <-> E. g e. ( ~P X i^i Fin ) ( F ` g ) = s ) )
9	6, 7, 8	sylancl 694	. . . . . . 7 |- ( C e. (ACS ` X ) -> ( s e. ( F " ( ~P X i^i Fin ) ) <-> E. g e. ( ~P X i^i Fin ) ( F ` g ) = s ) )
10		eqcom 2629	. . . . . . . 8 |- ( s = ( F ` g ) <-> ( F ` g ) = s )
11	10	rexbii 3041	. . . . . . 7 |- ( E. g e. ( ~P X i^i Fin ) s = ( F ` g ) <-> E. g e. ( ~P X i^i Fin ) ( F ` g ) = s )
12	9, 11	syl6rbbr 279	. . . . . 6 |- ( C e. (ACS ` X ) -> ( E. g e. ( ~P X i^i Fin ) s = ( F ` g ) <-> s e. ( F " ( ~P X i^i Fin ) ) ) )
13	12	ralbidv 2986	. . . . 5 |- ( C e. (ACS ` X ) -> ( A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) <-> A. s e. C s e. ( F " ( ~P X i^i Fin ) ) ) )
14		dfss3 3592	. . . . 5 |- ( C C_ ( F " ( ~P X i^i Fin ) ) <-> A. s e. C s e. ( F " ( ~P X i^i Fin ) ) )
15	13, 14	syl6bbr 278	. . . 4 |- ( C e. (ACS ` X ) -> ( A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) <-> C C_ ( F " ( ~P X i^i Fin ) ) ) )
16		imassrn 5477	. . . . . . 7 |- ( F " ( ~P X i^i Fin ) ) C_ ran F
17		frn 6053	. . . . . . . 8 |- ( F : ~P X --> C -> ran F C_ C )
18	3, 4, 17	3syl 18	. . . . . . 7 |- ( C e. (ACS ` X ) -> ran F C_ C )
19	16, 18	syl5ss 3614	. . . . . 6 |- ( C e. (ACS ` X ) -> ( F " ( ~P X i^i Fin ) ) C_ C )
20	19	biantrurd 529	. . . . 5 |- ( C e. (ACS ` X ) -> ( C C_ ( F " ( ~P X i^i Fin ) ) <-> ( ( F " ( ~P X i^i Fin ) ) C_ C /\ C C_ ( F " ( ~P X i^i Fin ) ) ) ) )
21		eqss 3618	. . . . 5 |- ( ( F " ( ~P X i^i Fin ) ) = C <-> ( ( F " ( ~P X i^i Fin ) ) C_ C /\ C C_ ( F " ( ~P X i^i Fin ) ) ) )
22	20, 21	syl6bbr 278	. . . 4 |- ( C e. (ACS ` X ) -> ( C C_ ( F " ( ~P X i^i Fin ) ) <-> ( F " ( ~P X i^i Fin ) ) = C ) )
23	15, 22	bitrd 268	. . 3 |- ( C e. (ACS ` X ) -> ( A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) <-> ( F " ( ~P X i^i Fin ) ) = C ) )
24	23	pm5.32i 669	. 2 |- ( ( C e. (ACS ` X ) /\ A. s e. C E. g e. ( ~P X i^i Fin ) s = ( F ` g ) ) <-> ( C e. (ACS ` X ) /\ ( F " ( ~P X i^i Fin ) ) = C ) )
25	2, 24	bitri 264	1 |- ( C e. (NoeACS ` X ) <-> ( C e. (ACS ` X ) /\ ( F " ( ~P X i^i Fin ) ) = C ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   Fincfn 7955  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245  NoeACScnacs 37265

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  df-acs 16249  df-nacs 37266

This theorem is referenced by:  nacsacs  37272