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Theorem isbnd 26607
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 5571 . 2  |-  ( M  e.  ( Bnd `  X
)  ->  X  e.  _V )
2 elfvex 5571 . . 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 5541 . . . . . 6  |-  ( y  =  X  ->  ( Met `  y )  =  ( Met `  X
) )
5 eqeq1 2302 . . . . . . . 8  |-  ( y  =  X  ->  (
y  =  ( x ( ball `  m
) r )  <->  X  =  ( x ( ball `  m ) r ) ) )
65rexbidv 2577 . . . . . . 7  |-  ( y  =  X  ->  ( E. r  e.  RR+  y  =  ( x (
ball `  m )
r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) ) )
76raleqbi1dv 2757 . . . . . 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 2796 . . . . 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 26606 . . . . 5  |-  Bnd  =  ( y  e.  _V  |->  { m  e.  ( Met `  y )  | 
A. x  e.  y  E. r  e.  RR+  y  =  ( x
( ball `  m )
r ) } )
10 fvex 5555 . . . . . 6  |-  ( Met `  X )  e.  _V
1110rabex 4181 . . . . 5  |-  { m  e.  ( Met `  X
)  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) }  e.  _V
128, 9, 11fvmpt 5618 . . . 4  |-  ( X  e.  _V  ->  ( Bnd `  X )  =  { m  e.  ( Met `  X )  |  A. x  e.  X  E. r  e.  RR+  X  =  ( x ( ball `  m
) r ) } )
1312eleq2d 2363 . . 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 5541 . . . . . . . 8  |-  ( m  =  M  ->  ( ball `  m )  =  ( ball `  M
) )
1514oveqd 5891 . . . . . . 7  |-  ( m  =  M  ->  (
x ( ball `  m
) r )  =  ( x ( ball `  M ) r ) )
1615eqeq2d 2307 . . . . . 6  |-  ( m  =  M  ->  ( X  =  ( x
( ball `  m )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
1716rexbidv 2577 . . . . 5  |-  ( m  =  M  ->  ( E. r  e.  RR+  X  =  ( x ( ball `  m ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
1817ralbidv 2576 . . . 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 2936 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   ` cfv 5271  (class class class)co 5874   RR+crp 10370   Metcme 16386   ballcbl 16387   Bndcbnd 26594
This theorem is referenced by:  bndmet  26608  isbndx  26609  isbnd3  26611  bndss  26613  totbndbnd  26616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-bnd 26606
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