Theorem bndmet 37113

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.

Step	Hyp	Ref	Expression
1		isbnd 37112	. 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑦)))
2	1	simplbi 497	1 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  ‘cfv 6543  (class class class)co 7412  ℝ⁺crp 12981  Metcmet 21219  ballcbl 21220  Bndcbnd 37099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-bnd 37111

This theorem is referenced by:  isbnd3  37116  equivbnd  37122  bnd2lem  37123  equivbnd2  37124  prdsbnd  37125  prdsbnd2  37127