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Theorem isbnd 30171
Description: The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
isbnd  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
Distinct variable groups:    x, r, M    X, r, x

Proof of Theorem isbnd
Dummy variables  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5898 . 2  |-  ( M  e.  ( Bnd `  X
)  ->  X  e.  _V )
2 elfvex 5898 . . 3  |-  ( M  e.  ( Met `  X
)  ->  X  e.  _V )
32adantr 465 . 2  |-  ( ( M  e.  ( Met `  X )  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) )  ->  X  e.  _V )
4 fveq2 5871 . . . . . 6  |-  ( y  =  X  ->  ( Met `  y )  =  ( Met `  X
) )
5 eqeq1 2471 . . . . . . . 8  |-  ( y  =  X  ->  (
y  =  ( x ( ball `  m
) r )  <->  X  =  ( x ( ball `  m ) r ) ) )
65rexbidv 2978 . . . . . . 7  |-  ( y  =  X  ->  ( E. r  e.  RR+  y  =  ( x (
ball `  m )
r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) ) )
76raleqbi1dv 3071 . . . . . 6  |-  ( y  =  X  ->  ( A. x  e.  y  E. r  e.  RR+  y  =  ( x (
ball `  m )
r )  <->  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) ) )
84, 7rabeqbidv 3113 . . . . 5  |-  ( y  =  X  ->  { m  e.  ( Met `  y
)  |  A. x  e.  y  E. r  e.  RR+  y  =  ( x ( ball `  m
) r ) }  =  { m  e.  ( Met `  X
)  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) } )
9 df-bnd 30170 . . . . 5  |-  Bnd  =  ( y  e.  _V  |->  { m  e.  ( Met `  y )  | 
A. x  e.  y  E. r  e.  RR+  y  =  ( x
( ball `  m )
r ) } )
10 fvex 5881 . . . . . 6  |-  ( Met `  X )  e.  _V
1110rabex 4603 . . . . 5  |-  { m  e.  ( Met `  X
)  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) }  e.  _V
128, 9, 11fvmpt 5956 . . . 4  |-  ( X  e.  _V  ->  ( Bnd `  X )  =  { m  e.  ( Met `  X )  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) } )
1312eleq2d 2537 . . 3  |-  ( X  e.  _V  ->  ( M  e.  ( Bnd `  X )  <->  M  e.  { m  e.  ( Met `  X )  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m ) r ) } ) )
14 fveq2 5871 . . . . . . . 8  |-  ( m  =  M  ->  ( ball `  m )  =  ( ball `  M
) )
1514oveqd 6311 . . . . . . 7  |-  ( m  =  M  ->  (
x ( ball `  m
) r )  =  ( x ( ball `  M ) r ) )
1615eqeq2d 2481 . . . . . 6  |-  ( m  =  M  ->  ( X  =  ( x
( ball `  m )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
1716rexbidv 2978 . . . . 5  |-  ( m  =  M  ->  ( E. r  e.  RR+  X  =  ( x ( ball `  m ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
1817ralbidv 2906 . . . 4  |-  ( m  =  M  ->  ( A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m ) r )  <->  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
1918elrab 3266 . . 3  |-  ( M  e.  { m  e.  ( Met `  X
)  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) }  <-> 
( M  e.  ( Met `  X )  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
2013, 19syl6bb 261 . 2  |-  ( X  e.  _V  ->  ( M  e.  ( Bnd `  X )  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) ) )
211, 3, 20pm5.21nii 353 1  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118   ` cfv 5593  (class class class)co 6294   RR+crp 11230   Metcme 18251   ballcbl 18252   Bndcbnd 30158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6297  df-bnd 30170
This theorem is referenced by:  bndmet  30172  isbndx  30173  isbnd3  30175  bndss  30177  totbndbnd  30180
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