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

Definition df-cls 16758
Description: Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 16774. (Contributed by NM, 3-Oct-2006.)
Assertion
Ref Expression
df-cls  |-  cls  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y } ) )
Distinct variable group:    x, j, y

Detailed syntax breakdown of Definition df-cls
StepHypRef Expression
1 ccl 16755 . 2  class  cls
2 vj . . 3  set  j
3 ctop 16631 . . 3  class  Top
4 vx . . . 4  set  x
52cv 1622 . . . . . 6  class  j
65cuni 3827 . . . . 5  class  U. j
76cpw 3625 . . . 4  class  ~P U. j
84cv 1622 . . . . . . 7  class  x
9 vy . . . . . . . 8  set  y
109cv 1622 . . . . . . 7  class  y
118, 10wss 3152 . . . . . 6  wff  x  C_  y
12 ccld 16753 . . . . . . 7  class  Clsd
135, 12cfv 5255 . . . . . 6  class  ( Clsd `  j )
1411, 9, 13crab 2547 . . . . 5  class  { y  e.  ( Clsd `  j
)  |  x  C_  y }
1514cint 3862 . . . 4  class  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y }
164, 7, 15cmpt 4077 . . 3  class  ( x  e.  ~P U. j  |-> 
|^| { y  e.  (
Clsd `  j )  |  x  C_  y } )
172, 3, 16cmpt 4077 . 2  class  ( j  e.  Top  |->  ( x  e.  ~P U. j  |-> 
|^| { y  e.  (
Clsd `  j )  |  x  C_  y } ) )
181, 17wceq 1623 1  wff  cls  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y } ) )
Colors of variables: wff set class
This definition is referenced by:  clsfval  16762
  Copyright terms: Public domain W3C validator