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

Definition df-ms 17880
Description: Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
df-ms  |-  MetSp  =  {
f  e.  * MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }

Detailed syntax breakdown of Definition df-ms
StepHypRef Expression
1 cmt 17877 . 2  class  MetSp
2 vf . . . . . . 7  set  f
32cv 1623 . . . . . 6  class  f
4 cds 13211 . . . . . 6  class  dist
53, 4cfv 5221 . . . . 5  class  ( dist `  f )
6 cbs 13142 . . . . . . 7  class  Base
73, 6cfv 5221 . . . . . 6  class  ( Base `  f )
87, 7cxp 4686 . . . . 5  class  ( (
Base `  f )  X.  ( Base `  f
) )
95, 8cres 4690 . . . 4  class  ( (
dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )
10 cme 16364 . . . . 5  class  Met
117, 10cfv 5221 . . . 4  class  ( Met `  ( Base `  f
) )
129, 11wcel 1685 . . 3  wff  ( (
dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f ) )
13 cxme 17876 . . 3  class  * MetSp
1412, 2, 13crab 2548 . 2  class  { f  e.  * MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }
151, 14wceq 1624 1  wff  MetSp  =  {
f  e.  * MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }
Colors of variables: wff set class
This definition is referenced by:  isms  17989
  Copyright terms: Public domain W3C validator