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

Definition df-mre 13504
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 16831) and vector spaces (lssmre 15739) 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 13508, mresspw 13510, mre1cl 13512 and mreintcl 13513 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13518); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13519. (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 13500 . 2  class Moore
2 vx . . 3  set  x
3 cvv 2801 . . 3  class  _V
4 vc . . . . . 6  set  c
52, 4wel 1697 . . . . 5  wff  x  e.  c
6 vs . . . . . . . . 9  set  s
76cv 1631 . . . . . . . 8  class  s
8 c0 3468 . . . . . . . 8  class  (/)
97, 8wne 2459 . . . . . . 7  wff  s  =/=  (/)
107cint 3878 . . . . . . . 8  class  |^| s
114cv 1631 . . . . . . . 8  class  c
1210, 11wcel 1696 . . . . . . 7  wff  |^| s  e.  c
139, 12wi 4 . . . . . 6  wff  ( s  =/=  (/)  ->  |^| s  e.  c )
1411cpw 3638 . . . . . 6  class  ~P c
1513, 6, 14wral 2556 . . . . 5  wff  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c )
165, 15wa 358 . . . 4  wff  ( x  e.  c  /\  A. s  e.  ~P  c
( s  =/=  (/)  ->  |^| s  e.  c ) )
172cv 1631 . . . . . 6  class  x
1817cpw 3638 . . . . 5  class  ~P x
1918cpw 3638 . . . 4  class  ~P ~P x
2016, 4, 19crab 2560 . . 3  class  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
212, 3, 20cmpt 4093 . 2  class  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
221, 21wceq 1632 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  13508  fnmre  13509
  Copyright terms: Public domain W3C validator