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Definition df-ms 20949
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 20946 . 2  class  MetSp
2 vf . . . . . . 7  setvar  f
32cv 1394 . . . . . 6  class  f
4 cds 14720 . . . . . 6  class  dist
53, 4cfv 5594 . . . . 5  class  ( dist `  f )
6 cbs 14643 . . . . . . 7  class  Base
73, 6cfv 5594 . . . . . 6  class  ( Base `  f )
87, 7cxp 5006 . . . . 5  class  ( (
Base `  f )  X.  ( Base `  f
) )
95, 8cres 5010 . . . 4  class  ( (
dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )
10 cme 18530 . . . . 5  class  Met
117, 10cfv 5594 . . . 4  class  ( Met `  ( Base `  f
) )
129, 11wcel 1819 . . 3  wff  ( (
dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f ) )
13 cxme 20945 . . 3  class  *MetSp
1412, 2, 13crab 2811 . 2  class  { f  e.  *MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }
151, 14wceq 1395 1  wff  MetSp  =  {
f  e.  *MetSp  |  ( ( dist `  f
)  |`  ( ( Base `  f )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f
) ) }
Colors of variables: wff setvar class
This definition is referenced by:  isms  21077
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