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Definition df-metu 18532
Description: Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
df-metu  |- metUnif  =  ( d  e.  U. ran PsMet  |->  ( ( dom  dom  d  X.  dom  dom  d
) filGen ran  ( a  e.  RR+  |->  ( `' d
" ( 0 [,) a ) ) ) ) )
Distinct variable group:    a, d

Detailed syntax breakdown of Definition df-metu
StepHypRef Expression
1 cmetu 18523 . 2  class metUnif
2 vd . . 3  setvar  d
3 cpsmet 18515 . . . . 5  class PsMet
43crn 4914 . . . 4  class  ran PsMet
54cuni 4163 . . 3  class  U. ran PsMet
62cv 1398 . . . . . . 7  class  d
76cdm 4913 . . . . . 6  class  dom  d
87cdm 4913 . . . . 5  class  dom  dom  d
98, 8cxp 4911 . . . 4  class  ( dom 
dom  d  X.  dom  dom  d )
10 va . . . . . 6  setvar  a
11 crp 11139 . . . . . 6  class  RR+
126ccnv 4912 . . . . . . 7  class  `' d
13 cc0 9403 . . . . . . . 8  class  0
1410cv 1398 . . . . . . . 8  class  a
15 cico 11452 . . . . . . . 8  class  [,)
1613, 14, 15co 6196 . . . . . . 7  class  ( 0 [,) a )
1712, 16cima 4916 . . . . . 6  class  ( `' d " ( 0 [,) a ) )
1810, 11, 17cmpt 4425 . . . . 5  class  ( a  e.  RR+  |->  ( `' d " ( 0 [,) a ) ) )
1918crn 4914 . . . 4  class  ran  (
a  e.  RR+  |->  ( `' d " ( 0 [,) a ) ) )
20 cfg 18520 . . . 4  class  filGen
219, 19, 20co 6196 . . 3  class  ( ( dom  dom  d  X.  dom  dom  d ) filGen ran  ( a  e.  RR+  |->  ( `' d " (
0 [,) a ) ) ) )
222, 5, 21cmpt 4425 . 2  class  ( d  e.  U. ran PsMet  |->  ( ( dom  dom  d  X.  dom  dom  d ) filGen ran  ( a  e.  RR+  |->  ( `' d " (
0 [,) a ) ) ) ) )
231, 22wceq 1399 1  wff metUnif  =  ( d  e.  U. ran PsMet  |->  ( ( dom  dom  d  X.  dom  dom  d
) filGen ran  ( a  e.  RR+  |->  ( `' d
" ( 0 [,) a ) ) ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  metuval  21138
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