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Definition df-mre 13842
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 17173) and vector spaces (lssmre 16073) 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 13846, mresspw 13848, mre1cl 13850 and mreintcl 13851 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13856); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13857. (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 13838 . 2  class Moore
2 vx . . 3  set  x
3 cvv 2962 . . 3  class  _V
4 vc . . . . . 6  set  c
52, 4wel 1728 . . . . 5  wff  x  e.  c
6 vs . . . . . . . . 9  set  s
76cv 1652 . . . . . . . 8  class  s
8 c0 3613 . . . . . . . 8  class  (/)
97, 8wne 2605 . . . . . . 7  wff  s  =/=  (/)
107cint 4074 . . . . . . . 8  class  |^| s
114cv 1652 . . . . . . . 8  class  c
1210, 11wcel 1727 . . . . . . 7  wff  |^| s  e.  c
139, 12wi 4 . . . . . 6  wff  ( s  =/=  (/)  ->  |^| s  e.  c )
1411cpw 3823 . . . . . 6  class  ~P c
1513, 6, 14wral 2711 . . . . 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 1652 . . . . . 6  class  x
1817cpw 3823 . . . . 5  class  ~P x
1918cpw 3823 . . . 4  class  ~P ~P x
2016, 4, 19crab 2715 . . 3  class  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
212, 3, 20cmpt 4291 . 2  class  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
221, 21wceq 1653 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  13846  fnmre  13847
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