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Theorem bndmet 33251
 Description: A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
bndmet (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Proof of Theorem bndmet
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbnd 33250 . 2 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑦 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑦)))
21simplbi 476 1 (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ∀wral 2908  ∃wrex 2909  ‘cfv 5857  (class class class)co 6615  ℝ+crp 11792  Metcme 19672  ballcbl 19673  Bndcbnd 33237 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-bnd 33249 This theorem is referenced by:  isbnd3  33254  equivbnd  33260  bnd2lem  33261  equivbnd2  33262  prdsbnd  33263  prdsbnd2  33265
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