Theorem ismri 16291

Description: Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)

Hypotheses

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

Assertion

Ref	Expression
ismri	|- ( A e. (Moore ` X ) -> ( S e. I <-> ( S C_ X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )

Distinct variable groups:   x, A   x, S

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

Proof of Theorem ismri

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismri.1	. . . . 5 |- N = (mrCls ` A )
2		ismri.2	. . . . 5 |- I = (mrInd ` A )
3	1, 2	mrisval 16290	. . . 4 |- ( A e. (Moore ` X ) -> I = { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )
4	3	eleq2d 2687	. . 3 |- ( A e. (Moore ` X ) -> ( S e. I <-> S e. { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } ) )
5		difeq1 3721	. . . . . . . 8 |- ( s = S -> ( s \ { x } ) = ( S \ { x } ) )
6	5	fveq2d 6195	. . . . . . 7 |- ( s = S -> ( N ` ( s \ { x } ) ) = ( N ` ( S \ { x } ) ) )
7	6	eleq2d 2687	. . . . . 6 |- ( s = S -> ( x e. ( N ` ( s \ { x } ) ) <-> x e. ( N ` ( S \ { x } ) ) ) )
8	7	notbid 308	. . . . 5 |- ( s = S -> ( -. x e. ( N ` ( s \ { x } ) ) <-> -. x e. ( N ` ( S \ { x } ) ) ) )
9	8	raleqbi1dv 3146	. . . 4 |- ( s = S -> ( A. x e. s -. x e. ( N ` ( s \ { x } ) ) <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
10	9	elrab 3363	. . 3 |- ( S e. { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } <-> ( S e. ~P X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
11	4, 10	syl6bb 276	. 2 |- ( A e. (Moore ` X ) -> ( S e. I <-> ( S e. ~P X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )
12		elfvex 6221	. . . 4 |- ( A e. (Moore ` X ) -> X e. _V )
13		elpw2g 4827	. . . 4 |- ( X e. _V -> ( S e. ~P X <-> S C_ X ) )
14	12, 13	syl 17	. . 3 |- ( A e. (Moore ` X ) -> ( S e. ~P X <-> S C_ X ) )
15	14	anbi1d 741	. 2 |- ( A e. (Moore ` X ) -> ( ( S e. ~P X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) <-> ( S C_ X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )
16	11, 15	bitrd 268	1 |- ( A e. (Moore ` X ) -> ( S e. I <-> ( S C_ X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   {csn 4177   ` 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:  ismri2  16292  mriss  16295  lbsacsbs  19156