Theorem mrisval 16290

Description: Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mrisval.1	|- N = (mrCls ` A )
mrisval.2	|- I = (mrInd ` A )

Assertion

Ref	Expression
mrisval	|- ( A e. (Moore ` X ) -> I = { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )

Distinct variable groups:   A, s, x   X, s

Allowed substitution hints:   I( x, s)   N( x, s)   X( x)

Proof of Theorem mrisval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrisval.2	. . 3 |- I = (mrInd ` A )
2		fvssunirn 6217	. . . . 5 |- (Moore ` X ) C_ U. ran Moore
3	2	sseli 3599	. . . 4 |- ( A e. (Moore ` X ) -> A e. U. ran Moore )
4		unieq 4444	. . . . . . 7 |- ( c = A -> U. c = U. A )
5	4	pweqd 4163	. . . . . 6 |- ( c = A -> ~P U. c = ~P U. A )
6		fveq2 6191	. . . . . . . . . . 11 |- ( c = A -> (mrCls ` c ) = (mrCls ` A ) )
7		mrisval.1	. . . . . . . . . . 11 |- N = (mrCls ` A )
8	6, 7	syl6eqr 2674	. . . . . . . . . 10 |- ( c = A -> (mrCls ` c ) = N )
9	8	fveq1d 6193	. . . . . . . . 9 |- ( c = A -> ( (mrCls ` c ) ` ( s \ { x } ) ) = ( N ` ( s \ { x } ) ) )
10	9	eleq2d 2687	. . . . . . . 8 |- ( c = A -> ( x e. ( (mrCls ` c ) ` ( s \ { x } ) ) <-> x e. ( N ` ( s \ { x } ) ) ) )
11	10	notbid 308	. . . . . . 7 |- ( c = A -> ( -. x e. ( (mrCls ` c ) ` ( s \ { x } ) ) <-> -. x e. ( N ` ( s \ { x } ) ) ) )
12	11	ralbidv 2986	. . . . . 6 |- ( c = A -> ( A. x e. s -. x e. ( (mrCls ` c ) ` ( s \ { x } ) ) <-> A. x e. s -. x e. ( N ` ( s \ { x } ) ) ) )
13	5, 12	rabeqbidv 3195	. . . . 5 |- ( c = A -> { s e. ~P U. c | A. x e. s -. x e. ( (mrCls ` c ) ` ( s \ { x } ) ) } = { s e. ~P U. A | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )
14		df-mri 16248	. . . . 5 |- mrInd = ( c e. U. ran Moore |-> { s e. ~P U. c | A. x e. s -. x e. ( (mrCls ` c ) ` ( s \ { x } ) ) } )
15		vuniex 6954	. . . . . . 7 |- U. c e. _V
16	15	pwex 4848	. . . . . 6 |- ~P U. c e. _V
17	16	rabex 4813	. . . . 5 |- { s e. ~P U. c | A. x e. s -. x e. ( (mrCls ` c ) ` ( s \ { x } ) ) } e. _V
18	13, 14, 17	fvmpt3i 6287	. . . 4 |- ( A e. U. ran Moore -> (mrInd ` A ) = { s e. ~P U. A | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )
19	3, 18	syl 17	. . 3 |- ( A e. (Moore ` X ) -> (mrInd ` A ) = { s e. ~P U. A | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )
20	1, 19	syl5eq 2668	. 2 |- ( A e. (Moore ` X ) -> I = { s e. ~P U. A | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )
21		mreuni 16260	. . . 4 |- ( A e. (Moore ` X ) -> U. A = X )
22	21	pweqd 4163	. . 3 |- ( A e. (Moore ` X ) -> ~P U. A = ~P X )
23	22	rabeqdv 3194	. 2 |- ( A e. (Moore ` X ) -> { s e. ~P U. A | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } = { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )
24	20, 23	eqtrd 2656	1 |- ( A e. (Moore ` X ) -> I = { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   ~Pcpw 4158   {csn 4177   U.cuni 4436   ran crn 5115   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243  mrIndcmri 16244

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-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-fv 5896  df-mre 16246  df-mri 16248

This theorem is referenced by:  ismri  16291