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Metamath Proof Explorer

Theorem ismri

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

Ref Expression
Hypotheses ismri.1 
|- N = ( mrCls ` A )
ismri.2 
|- I = ( mrInd ` A )
Assertion ismri 
|- ( A e. ( Moore ` X ) -> ( S e. I <-> ( S C_ X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )

Proof

Step Hyp Ref Expression
1 ismri.1 
 |-  N = ( mrCls ` A )
2 ismri.2 
 |-  I = ( mrInd ` A )
3 1 2 mrisval 
 |-  ( A e. ( Moore ` X ) -> I = { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )
4 3 eleq2d 
 |-  ( A e. ( Moore ` X ) -> ( S e. I <-> S e. { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } ) )
5 difeq1 
 |-  ( s = S -> ( s \ { x } ) = ( S \ { x } ) )
6 5 fveq2d 
 |-  ( s = S -> ( N ` ( s \ { x } ) ) = ( N ` ( S \ { x } ) ) )
7 6 eleq2d 
 |-  ( s = S -> ( x e. ( N ` ( s \ { x } ) ) <-> x e. ( N ` ( S \ { x } ) ) ) )
8 7 notbid 
 |-  ( s = S -> ( -. x e. ( N ` ( s \ { x } ) ) <-> -. x e. ( N ` ( S \ { x } ) ) ) )
9 8 raleqbi1dv 
 |-  ( s = S -> ( A. x e. s -. x e. ( N ` ( s \ { x } ) ) <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
10 9 elrab 
 |-  ( 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 bitrdi 
 |-  ( A e. ( Moore ` X ) -> ( S e. I <-> ( S e. ~P X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )
12 elfvex 
 |-  ( A e. ( Moore ` X ) -> X e. _V )
13 elpw2g 
 |-  ( X e. _V -> ( S e. ~P X <-> S C_ X ) )
14 12 13 syl 
 |-  ( A e. ( Moore ` X ) -> ( S e. ~P X <-> S C_ X ) )
15 14 anbi1d 
 |-  ( 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 
 |-  ( A e. ( Moore ` X ) -> ( S e. I <-> ( S C_ X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )