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Theorem isbnd 28684
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 5722 . 2  |-  ( M  e.  ( Bnd `  X
)  ->  X  e.  _V )
2 elfvex 5722 . . 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 5696 . . . . . 6  |-  ( y  =  X  ->  ( Met `  y )  =  ( Met `  X
) )
5 eqeq1 2449 . . . . . . . 8  |-  ( y  =  X  ->  (
y  =  ( x ( ball `  m
) r )  <->  X  =  ( x ( ball `  m ) r ) ) )
65rexbidv 2741 . . . . . . 7  |-  ( y  =  X  ->  ( E. r  e.  RR+  y  =  ( x (
ball `  m )
r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) ) )
76raleqbi1dv 2930 . . . . . 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 2972 . . . . 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 28683 . . . . 5  |-  Bnd  =  ( y  e.  _V  |->  { m  e.  ( Met `  y )  | 
A. x  e.  y  E. r  e.  RR+  y  =  ( x
( ball `  m )
r ) } )
10 fvex 5706 . . . . . 6  |-  ( Met `  X )  e.  _V
1110rabex 4448 . . . . 5  |-  { m  e.  ( Met `  X
)  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) }  e.  _V
128, 9, 11fvmpt 5779 . . . 4  |-  ( X  e.  _V  ->  ( Bnd `  X )  =  { m  e.  ( Met `  X )  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) } )
1312eleq2d 2510 . . 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 5696 . . . . . . . 8  |-  ( m  =  M  ->  ( ball `  m )  =  ( ball `  M
) )
1514oveqd 6113 . . . . . . 7  |-  ( m  =  M  ->  (
x ( ball `  m
) r )  =  ( x ( ball `  M ) r ) )
1615eqeq2d 2454 . . . . . 6  |-  ( m  =  M  ->  ( X  =  ( x
( ball `  m )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
1716rexbidv 2741 . . . . 5  |-  ( m  =  M  ->  ( E. r  e.  RR+  X  =  ( x ( ball `  m ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
1817ralbidv 2740 . . . 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 3122 . . 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 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977   ` cfv 5423  (class class class)co 6096   RR+crp 10996   Metcme 17807   ballcbl 17808   Bndcbnd 28671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-bnd 28683
This theorem is referenced by:  bndmet  28685  isbndx  28686  isbnd3  28688  bndss  28690  totbndbnd  28693
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