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Definition df-mre 13450
Description: Define a Moore collection, which is a family of subsets of a base set which preserve arbitrary intersection. Elements of a Moore collection are termed closed; Moore collections generalize the notion of closedness from topologies (cldmre 16777) and vector spaces (lssmre 15685) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78. A Moore collection may also be called a closure system (Section 0.6 in [Gratzer] p. 23.) The name Moore collection is after Eliakim Hastings Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.

See ismre 13454, mresspw 13456, mre1cl 13458 and mreintcl 13459 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13464); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13465. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

Assertion
Ref Expression
df-mre  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
Distinct variable group:    s, c, x

Detailed syntax breakdown of Definition df-mre
StepHypRef Expression
1 cmre 13446 . 2  class Moore
2 vx . . 3  set  x
3 cvv 2763 . . 3  class  _V
4 vc . . . . . 6  set  c
52, 4wel 1622 . . . . 5  wff  x  e.  c
6 vs . . . . . . . . 9  set  s
76cv 1618 . . . . . . . 8  class  s
8 c0 3430 . . . . . . . 8  class  (/)
97, 8wne 2421 . . . . . . 7  wff  s  =/=  (/)
107cint 3836 . . . . . . . 8  class  |^| s
114cv 1618 . . . . . . . 8  class  c
1210, 11wcel 1621 . . . . . . 7  wff  |^| s  e.  c
139, 12wi 6 . . . . . 6  wff  ( s  =/=  (/)  ->  |^| s  e.  c )
1411cpw 3599 . . . . . 6  class  ~P c
1513, 6, 14wral 2518 . . . . 5  wff  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c )
165, 15wa 360 . . . 4  wff  ( x  e.  c  /\  A. s  e.  ~P  c
( s  =/=  (/)  ->  |^| s  e.  c ) )
172cv 1618 . . . . . 6  class  x
1817cpw 3599 . . . . 5  class  ~P x
1918cpw 3599 . . . 4  class  ~P ~P x
2016, 4, 19crab 2522 . . 3  class  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
212, 3, 20cmpt 4051 . 2  class  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
221, 21wceq 1619 1  wff Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
Colors of variables: wff set class
This definition is referenced by:  ismre  13454  fnmre  13455
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