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Theorem bndmet 30211
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 30210 . 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 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   ` cfv 5594  (class class class)co 6295   RR+crp 11232   Metcme 18272   ballcbl 18273   Bndcbnd 30197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-bnd 30209
This theorem is referenced by:  isbnd3  30214  equivbnd  30220  bnd2lem  30221  equivbnd2  30222  prdsbnd  30223  prdsbnd2  30225
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