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Theorem isbnd 26504
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 5555 . 2  |-  ( M  e.  ( Bnd `  X
)  ->  X  e.  _V )
2 elfvex 5555 . . 3  |-  ( M  e.  ( Met `  X
)  ->  X  e.  _V )
32adantr 451 . 2  |-  ( ( M  e.  ( Met `  X )  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) )  ->  X  e.  _V )
4 fveq2 5525 . . . . . 6  |-  ( y  =  X  ->  ( Met `  y )  =  ( Met `  X
) )
5 eqeq1 2289 . . . . . . . 8  |-  ( y  =  X  ->  (
y  =  ( x ( ball `  m
) r )  <->  X  =  ( x ( ball `  m ) r ) ) )
65rexbidv 2564 . . . . . . 7  |-  ( y  =  X  ->  ( E. r  e.  RR+  y  =  ( x (
ball `  m )
r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) ) )
76raleqbi1dv 2744 . . . . . 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 2783 . . . . 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 26503 . . . . 5  |-  Bnd  =  ( y  e.  _V  |->  { m  e.  ( Met `  y )  | 
A. x  e.  y  E. r  e.  RR+  y  =  ( x
( ball `  m )
r ) } )
10 fvex 5539 . . . . . 6  |-  ( Met `  X )  e.  _V
1110rabex 4165 . . . . 5  |-  { m  e.  ( Met `  X
)  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) }  e.  _V
128, 9, 11fvmpt 5602 . . . 4  |-  ( X  e.  _V  ->  ( Bnd `  X )  =  { m  e.  ( Met `  X )  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) } )
1312eleq2d 2350 . . 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 5525 . . . . . . . 8  |-  ( m  =  M  ->  ( ball `  m )  =  ( ball `  M
) )
1514oveqd 5875 . . . . . . 7  |-  ( m  =  M  ->  (
x ( ball `  m
) r )  =  ( x ( ball `  M ) r ) )
1615eqeq2d 2294 . . . . . 6  |-  ( m  =  M  ->  ( X  =  ( x
( ball `  m )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
1716rexbidv 2564 . . . . 5  |-  ( m  =  M  ->  ( E. r  e.  RR+  X  =  ( x ( ball `  m ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
1817ralbidv 2563 . . . 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 2923 . . 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 252 . 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 342 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 set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788   ` cfv 5255  (class class class)co 5858   RR+crp 10354   Metcme 16370   ballcbl 16371   Bndcbnd 26491
This theorem is referenced by:  bndmet  26505  isbndx  26506  isbnd3  26508  bndss  26510  totbndbnd  26513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-bnd 26503
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