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Definition df-bnd 26606
Description: Define the class of bounded metrics. A metric space is bounded iff it can be covered by a single ball. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
df-bnd  |-  Bnd  =  ( x  e.  _V  |->  { m  e.  ( Met `  x )  | 
A. y  e.  x  E. r  e.  RR+  x  =  ( y (
ball `  m )
r ) } )
Distinct variable group:    m, r, x, y

Detailed syntax breakdown of Definition df-bnd
StepHypRef Expression
1 cbnd 26594 . 2  class  Bnd
2 vx . . 3  set  x
3 cvv 2801 . . 3  class  _V
42cv 1631 . . . . . . 7  class  x
5 vy . . . . . . . . 9  set  y
65cv 1631 . . . . . . . 8  class  y
7 vr . . . . . . . . 9  set  r
87cv 1631 . . . . . . . 8  class  r
9 vm . . . . . . . . . 10  set  m
109cv 1631 . . . . . . . . 9  class  m
11 cbl 16387 . . . . . . . . 9  class  ball
1210, 11cfv 5271 . . . . . . . 8  class  ( ball `  m )
136, 8, 12co 5874 . . . . . . 7  class  ( y ( ball `  m
) r )
144, 13wceq 1632 . . . . . 6  wff  x  =  ( y ( ball `  m ) r )
15 crp 10370 . . . . . 6  class  RR+
1614, 7, 15wrex 2557 . . . . 5  wff  E. r  e.  RR+  x  =  ( y ( ball `  m
) r )
1716, 5, 4wral 2556 . . . 4  wff  A. y  e.  x  E. r  e.  RR+  x  =  ( y ( ball `  m
) r )
18 cme 16386 . . . . 5  class  Met
194, 18cfv 5271 . . . 4  class  ( Met `  x )
2017, 9, 19crab 2560 . . 3  class  { m  e.  ( Met `  x
)  |  A. y  e.  x  E. r  e.  RR+  x  =  ( y ( ball `  m
) r ) }
212, 3, 20cmpt 4093 . 2  class  ( x  e.  _V  |->  { m  e.  ( Met `  x
)  |  A. y  e.  x  E. r  e.  RR+  x  =  ( y ( ball `  m
) r ) } )
221, 21wceq 1632 1  wff  Bnd  =  ( x  e.  _V  |->  { m  e.  ( Met `  x )  | 
A. y  e.  x  E. r  e.  RR+  x  =  ( y (
ball `  m )
r ) } )
Colors of variables: wff set class
This definition is referenced by:  isbnd  26607
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