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Theorem bndmet 35595
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 35594 . 2 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑦 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑦)))
21simplbi 501 1 (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  wral 3054  wrex 3055  cfv 6350  (class class class)co 7183  +crp 12485  Metcmet 20216  ballcbl 20217  Bndcbnd 35581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-iota 6308  df-fun 6352  df-fv 6358  df-ov 7186  df-bnd 35593
This theorem is referenced by:  isbnd3  35598  equivbnd  35604  bnd2lem  35605  equivbnd2  35606  prdsbnd  35607  prdsbnd2  35609
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