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Definition df-bl 16471
Description: Define the metric space ball function. See blval 18044 for its value. (Contributed by NM, 30-Aug-2006.)
Assertion
Ref Expression
df-bl  |-  ball  =  ( d  e.  U. ran  * Met  |->  ( x  e.  dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  <  z } ) )
Distinct variable group:    x, d, y, z

Detailed syntax breakdown of Definition df-bl
StepHypRef Expression
1 cbl 16464 . 2  class  ball
2 vd . . 3  set  d
3 cxmt 16462 . . . . 5  class  * Met
43crn 4769 . . . 4  class  ran  * Met
54cuni 3906 . . 3  class  U. ran  * Met
6 vx . . . 4  set  x
7 vz . . . 4  set  z
82cv 1641 . . . . . 6  class  d
98cdm 4768 . . . . 5  class  dom  d
109cdm 4768 . . . 4  class  dom  dom  d
11 cxr 8953 . . . 4  class  RR*
126cv 1641 . . . . . . 7  class  x
13 vy . . . . . . . 8  set  y
1413cv 1641 . . . . . . 7  class  y
1512, 14, 8co 5942 . . . . . 6  class  ( x d y )
167cv 1641 . . . . . 6  class  z
17 clt 8954 . . . . . 6  class  <
1815, 16, 17wbr 4102 . . . . 5  wff  ( x d y )  < 
z
1918, 13, 10crab 2623 . . . 4  class  { y  e.  dom  dom  d  |  ( x d y )  <  z }
206, 7, 10, 11, 19cmpt2 5944 . . 3  class  ( x  e.  dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  <  z } )
212, 5, 20cmpt 4156 . 2  class  ( d  e.  U. ran  * Met  |->  ( x  e. 
dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  <  z } ) )
221, 21wceq 1642 1  wff  ball  =  ( d  e.  U. ran  * Met  |->  ( x  e.  dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  <  z } ) )
Colors of variables: wff set class
This definition is referenced by:  blfval  18043
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