Theorem bndmet 32750

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 32749	. 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑦)))
2	1	simplbi 475	1 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ‘cfv 5804  (class class class)co 6549  ℝ⁺crp 11708  Metcme 19553  ballcbl 19554  Bndcbnd 32736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-bnd 32748

This theorem is referenced by:  isbnd3  32753  equivbnd  32759  bnd2lem  32760  equivbnd2  32761  prdsbnd  32762  prdsbnd2  32764