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Theorem bndmet 28685
Description: A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
bndmet  |-  ( M  e.  ( Bnd `  X
)  ->  M  e.  ( Met `  X ) )

Proof of Theorem bndmet
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbnd 28684 . 2  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. x  e.  X  E. y  e.  RR+  X  =  ( x ( ball `  M
) y ) ) )
21simplbi 460 1  |-  ( M  e.  ( Bnd `  X
)  ->  M  e.  ( Met `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   ` 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:  isbnd3  28688  equivbnd  28694  bnd2lem  28695  equivbnd2  28696  prdsbnd  28697  prdsbnd2  28699
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