Theorem totbndmet

Description: The predicate "totally bounded" implies M is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Assertion	totbndmet	⊢ ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) → 𝑀 ∈ ( Met ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		istotbnd	⊢ ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) ↔ ( 𝑀 ∈ ( Met ‘ 𝑋 ) ∧ ∀ 𝑑 ∈ ℝ+ ∃ 𝑣 ∈ Fin ( ∪ 𝑣 = 𝑋 ∧ ∀ 𝑏 ∈ 𝑣 ∃ 𝑥 ∈ 𝑋 𝑏 = ( 𝑥 ( ball ‘ 𝑀 ) 𝑑 ) ) ) )
2	1	simplbi	⊢ ( 𝑀 ∈ ( TotBnd ‘ 𝑋 ) → 𝑀 ∈ ( Met ‘ 𝑋 ) )