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Definition df-bl 16702
Description: Define the metric space ball function. See blval 18421 for its value. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
df-bl  |-  ball  =  ( d  e.  _V  |->  ( 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 16693 . 2  class  ball
2 vd . . 3  set  d
3 cvv 2958 . . 3  class  _V
4 vx . . . 4  set  x
5 vz . . . 4  set  z
62cv 1652 . . . . . 6  class  d
76cdm 4881 . . . . 5  class  dom  d
87cdm 4881 . . . 4  class  dom  dom  d
9 cxr 9124 . . . 4  class  RR*
104cv 1652 . . . . . . 7  class  x
11 vy . . . . . . . 8  set  y
1211cv 1652 . . . . . . 7  class  y
1310, 12, 6co 6084 . . . . . 6  class  ( x d y )
145cv 1652 . . . . . 6  class  z
15 clt 9125 . . . . . 6  class  <
1613, 14, 15wbr 4215 . . . . 5  wff  ( x d y )  < 
z
1716, 11, 8crab 2711 . . . 4  class  { y  e.  dom  dom  d  |  ( x d y )  <  z }
184, 5, 8, 9, 17cmpt2 6086 . . 3  class  ( x  e.  dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  <  z } )
192, 3, 18cmpt 4269 . 2  class  ( d  e.  _V  |->  ( x  e.  dom  dom  d ,  z  e.  RR*  |->  { y  e.  dom  dom  d  |  ( x d y )  <  z } ) )
201, 19wceq 1653 1  wff  ball  =  ( d  e.  _V  |->  ( 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:  blfvalps  18418
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