Definition df-watsN 35276

Description: Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom" d. These are all atoms not in the polarity of { d } ), which is the hyperplane determined by d. Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)

Assertion

Ref	Expression
df-watsN	|- WAtoms = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) ) ) )

Distinct variable group:   k, d

Detailed syntax breakdown of Definition df-watsN

Step	Hyp	Ref	Expression
1		cwpointsN 35272	. 2 class WAtoms
2		vk	. . 3 setvar k
3		cvv 3200	. . 3 class _V
4		vd	. . . 4 setvar d
5	2	cv 1482	. . . . 5 class k
6		catm 34550	. . . . 5 class Atoms
7	5, 6	cfv 5888	. . . 4 class ( Atoms ` k )
8	4	cv 1482	. . . . . . 7 class d
9	8	csn 4177	. . . . . 6 class { d }
10		cpolN 35188	. . . . . . 7 class _|_P
11	5, 10	cfv 5888	. . . . . 6 class ( _|_P ` k )
12	9, 11	cfv 5888	. . . . 5 class ( ( _|_P ` k ) ` { d } )
13	7, 12	cdif 3571	. . . 4 class ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) )
14	4, 7, 13	cmpt 4729	. . 3 class ( d e. ( Atoms ` k ) |-> ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) ) )
15	2, 3, 14	cmpt 4729	. 2 class ( k e. _V |-> ( d e. ( Atoms ` k ) |-> ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) ) ) )
16	1, 15	wceq 1483	1 wff WAtoms = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) ) ) )

This definition is referenced by:  watfvalN  35278