Theorem bndmet

Description: A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015)

	Ref	Expression
Assertion	bndmet	⊢ ( 𝑀 ∈ ( Bnd ‘ 𝑋 ) → 𝑀 ∈ ( Met ‘ 𝑋 ) )

Proof

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