Theorem mrefg3 37271

Description: Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Hypothesis

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

Assertion

Ref	Expression
mrefg3	|- ( ( C e. (Moore ` X ) /\ S e. C ) -> ( E. g e. ( ~P X i^i Fin ) S = ( F ` g ) <-> E. g e. ( ~P S i^i Fin ) S C_ ( F ` g ) ) )

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

Proof of Theorem mrefg3

Step	Hyp	Ref	Expression
1		isnacs.f	. . . 4 |- F = (mrCls ` C )
2	1	mrefg2 37270	. . 3 |- ( C e. (Moore ` X ) -> ( E. g e. ( ~P X i^i Fin ) S = ( F ` g ) <-> E. g e. ( ~P S i^i Fin ) S = ( F ` g ) ) )
3	2	adantr 481	. 2 |- ( ( C e. (Moore ` X ) /\ S e. C ) -> ( E. g e. ( ~P X i^i Fin ) S = ( F ` g ) <-> E. g e. ( ~P S i^i Fin ) S = ( F ` g ) ) )
4		simpll 790	. . . . . 6 |- ( ( ( C e. (Moore ` X ) /\ S e. C ) /\ g e. ( ~P S i^i Fin ) ) -> C e. (Moore ` X ) )
5		inss1 3833	. . . . . . . . 9 |- ( ~P S i^i Fin ) C_ ~P S
6	5	sseli 3599	. . . . . . . 8 |- ( g e. ( ~P S i^i Fin ) -> g e. ~P S )
7	6	elpwid 4170	. . . . . . 7 |- ( g e. ( ~P S i^i Fin ) -> g C_ S )
8	7	adantl 482	. . . . . 6 |- ( ( ( C e. (Moore ` X ) /\ S e. C ) /\ g e. ( ~P S i^i Fin ) ) -> g C_ S )
9		simplr 792	. . . . . 6 |- ( ( ( C e. (Moore ` X ) /\ S e. C ) /\ g e. ( ~P S i^i Fin ) ) -> S e. C )
10	1	mrcsscl 16280	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ g C_ S /\ S e. C ) -> ( F ` g ) C_ S )
11	4, 8, 9, 10	syl3anc 1326	. . . . 5 |- ( ( ( C e. (Moore ` X ) /\ S e. C ) /\ g e. ( ~P S i^i Fin ) ) -> ( F ` g ) C_ S )
12	11	biantrud 528	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ S e. C ) /\ g e. ( ~P S i^i Fin ) ) -> ( S C_ ( F ` g ) <-> ( S C_ ( F ` g ) /\ ( F ` g ) C_ S ) ) )
13		eqss 3618	. . . 4 |- ( S = ( F ` g ) <-> ( S C_ ( F ` g ) /\ ( F ` g ) C_ S ) )
14	12, 13	syl6rbbr 279	. . 3 |- ( ( ( C e. (Moore ` X ) /\ S e. C ) /\ g e. ( ~P S i^i Fin ) ) -> ( S = ( F ` g ) <-> S C_ ( F ` g ) ) )
15	14	rexbidva 3049	. 2 |- ( ( C e. (Moore ` X ) /\ S e. C ) -> ( E. g e. ( ~P S i^i Fin ) S = ( F ` g ) <-> E. g e. ( ~P S i^i Fin ) S C_ ( F ` g ) ) )
16	3, 15	bitrd 268	1 |- ( ( C e. (Moore ` X ) /\ S e. C ) -> ( E. g e. ( ~P X i^i Fin ) S = ( F ` g ) <-> E. g e. ( ~P S i^i Fin ) S C_ ( F ` g ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   ` cfv 5888   Fincfn 7955  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-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-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: (None)