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Definition df-psubclN 30124
Description: Define set of all closed projective subspaces, which are those sets of atoms that equal their double polarity. Based on definition in [Holland95] p. 223. (Contributed by NM, 23-Jan-2012.)
Assertion
Ref Expression
df-psubclN  |-  PSubCl  =  ( k  e.  _V  |->  { s  |  ( s 
C_  ( Atoms `  k
)  /\  ( ( _|_ P `  k ) `
 ( ( _|_
P `  k ) `  s ) )  =  s ) } )
Distinct variable group:    k, s

Detailed syntax breakdown of Definition df-psubclN
StepHypRef Expression
1 cpscN 30123 . 2  class  PSubCl
2 vk . . 3  set  k
3 cvv 2788 . . 3  class  _V
4 vs . . . . . . 7  set  s
54cv 1622 . . . . . 6  class  s
62cv 1622 . . . . . . 7  class  k
7 catm 29453 . . . . . . 7  class  Atoms
86, 7cfv 5255 . . . . . 6  class  ( Atoms `  k )
95, 8wss 3152 . . . . 5  wff  s  C_  ( Atoms `  k )
10 cpolN 30091 . . . . . . . . 9  class  _|_ P
116, 10cfv 5255 . . . . . . . 8  class  ( _|_
P `  k )
125, 11cfv 5255 . . . . . . 7  class  ( ( _|_ P `  k
) `  s )
1312, 11cfv 5255 . . . . . 6  class  ( ( _|_ P `  k
) `  ( ( _|_ P `  k ) `
 s ) )
1413, 5wceq 1623 . . . . 5  wff  ( ( _|_ P `  k
) `  ( ( _|_ P `  k ) `
 s ) )  =  s
159, 14wa 358 . . . 4  wff  ( s 
C_  ( Atoms `  k
)  /\  ( ( _|_ P `  k ) `
 ( ( _|_
P `  k ) `  s ) )  =  s )
1615, 4cab 2269 . . 3  class  { s  |  ( s  C_  ( Atoms `  k )  /\  ( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s ) }
172, 3, 16cmpt 4077 . 2  class  ( k  e.  _V  |->  { s  |  ( s  C_  ( Atoms `  k )  /\  ( ( _|_ P `  k ) `  (
( _|_ P `  k ) `  s
) )  =  s ) } )
181, 17wceq 1623 1  wff  PSubCl  =  ( k  e.  _V  |->  { s  |  ( s 
C_  ( Atoms `  k
)  /\  ( ( _|_ P `  k ) `
 ( ( _|_
P `  k ) `  s ) )  =  s ) } )
Colors of variables: wff set class
This definition is referenced by:  psubclsetN  30125
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