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Theorem isbnd 26491
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 5760 . 2  |-  ( M  e.  ( Bnd `  X
)  ->  X  e.  _V )
2 elfvex 5760 . . 3  |-  ( M  e.  ( Met `  X
)  ->  X  e.  _V )
32adantr 453 . 2  |-  ( ( M  e.  ( Met `  X )  /\  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) )  ->  X  e.  _V )
4 fveq2 5730 . . . . . 6  |-  ( y  =  X  ->  ( Met `  y )  =  ( Met `  X
) )
5 eqeq1 2444 . . . . . . . 8  |-  ( y  =  X  ->  (
y  =  ( x ( ball `  m
) r )  <->  X  =  ( x ( ball `  m ) r ) ) )
65rexbidv 2728 . . . . . . 7  |-  ( y  =  X  ->  ( E. r  e.  RR+  y  =  ( x (
ball `  m )
r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) ) )
76raleqbi1dv 2914 . . . . . 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 2953 . . . . 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 26490 . . . . 5  |-  Bnd  =  ( y  e.  _V  |->  { m  e.  ( Met `  y )  | 
A. x  e.  y  E. r  e.  RR+  y  =  ( x
( ball `  m )
r ) } )
10 fvex 5744 . . . . . 6  |-  ( Met `  X )  e.  _V
1110rabex 4356 . . . . 5  |-  { m  e.  ( Met `  X
)  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) }  e.  _V
128, 9, 11fvmpt 5808 . . . 4  |-  ( X  e.  _V  ->  ( Bnd `  X )  =  { m  e.  ( Met `  X )  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) } )
1312eleq2d 2505 . . 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 5730 . . . . . . . 8  |-  ( m  =  M  ->  ( ball `  m )  =  ( ball `  M
) )
1514oveqd 6100 . . . . . . 7  |-  ( m  =  M  ->  (
x ( ball `  m
) r )  =  ( x ( ball `  M ) r ) )
1615eqeq2d 2449 . . . . . 6  |-  ( m  =  M  ->  ( X  =  ( x
( ball `  m )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
1716rexbidv 2728 . . . . 5  |-  ( m  =  M  ->  ( E. r  e.  RR+  X  =  ( x ( ball `  m ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
1817ralbidv 2727 . . . 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 3094 . . 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 254 . 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 344 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958   ` cfv 5456  (class class class)co 6083   RR+crp 10614   Metcme 16689   ballcbl 16690   Bndcbnd 26478
This theorem is referenced by:  bndmet  26492  isbndx  26493  isbnd3  26495  bndss  26497  totbndbnd  26500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-bnd 26490
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