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Definition df-mre 13482
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 16809) and vector spaces (lssmre 15717) 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 13486, mresspw 13488, mre1cl 13490 and mreintcl 13491 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13496); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13497. (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 13478 . 2  class Moore
2 vx . . 3  set  x
3 cvv 2789 . . 3  class  _V
4 vc . . . . . 6  set  c
52, 4wel 1686 . . . . 5  wff  x  e.  c
6 vs . . . . . . . . 9  set  s
76cv 1623 . . . . . . . 8  class  s
8 c0 3456 . . . . . . . 8  class  (/)
97, 8wne 2447 . . . . . . 7  wff  s  =/=  (/)
107cint 3863 . . . . . . . 8  class  |^| s
114cv 1623 . . . . . . . 8  class  c
1210, 11wcel 1685 . . . . . . 7  wff  |^| s  e.  c
139, 12wi 6 . . . . . 6  wff  ( s  =/=  (/)  ->  |^| s  e.  c )
1411cpw 3626 . . . . . 6  class  ~P c
1513, 6, 14wral 2544 . . . . 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 1623 . . . . . 6  class  x
1817cpw 3626 . . . . 5  class  ~P x
1918cpw 3626 . . . 4  class  ~P ~P x
2016, 4, 19crab 2548 . . 3  class  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
212, 3, 20cmpt 4078 . 2  class  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
221, 21wceq 1624 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  13486  fnmre  13487
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