MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-mre Unicode version

Definition df-mre 13739
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 17066) and vector spaces (lssmre 15970) 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 13743, mresspw 13745, mre1cl 13747 and mreintcl 13748 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13753); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13754. (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 13735 . 2  class Moore
2 vx . . 3  set  x
3 cvv 2900 . . 3  class  _V
4 vc . . . . . 6  set  c
52, 4wel 1718 . . . . 5  wff  x  e.  c
6 vs . . . . . . . . 9  set  s
76cv 1648 . . . . . . . 8  class  s
8 c0 3572 . . . . . . . 8  class  (/)
97, 8wne 2551 . . . . . . 7  wff  s  =/=  (/)
107cint 3993 . . . . . . . 8  class  |^| s
114cv 1648 . . . . . . . 8  class  c
1210, 11wcel 1717 . . . . . . 7  wff  |^| s  e.  c
139, 12wi 4 . . . . . 6  wff  ( s  =/=  (/)  ->  |^| s  e.  c )
1411cpw 3743 . . . . . 6  class  ~P c
1513, 6, 14wral 2650 . . . . 5  wff  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c )
165, 15wa 359 . . . 4  wff  ( x  e.  c  /\  A. s  e.  ~P  c
( s  =/=  (/)  ->  |^| s  e.  c ) )
172cv 1648 . . . . . 6  class  x
1817cpw 3743 . . . . 5  class  ~P x
1918cpw 3743 . . . 4  class  ~P ~P x
2016, 4, 19crab 2654 . . 3  class  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
212, 3, 20cmpt 4208 . 2  class  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
221, 21wceq 1649 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  13743  fnmre  13744
  Copyright terms: Public domain W3C validator