Theorem msf

Description: The distance function of a metric space is a function into the real numbers. (Contributed by NM, 30-Aug-2006) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypotheses	msf.x	⊢ 𝑋 = ( Base ‘ 𝑀 )
	msf.d	⊢ 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) )
Assertion	msf	⊢ ( 𝑀 ∈ MetSp → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ )

Proof

Step	Hyp	Ref	Expression
1		msf.x	⊢ 𝑋 = ( Base ‘ 𝑀 )
2		msf.d	⊢ 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) )
3	1 2	msmet	⊢ ( 𝑀 ∈ MetSp → 𝐷 ∈ ( Met ‘ 𝑋 ) )
4		metf	⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ )
5	3 4	syl	⊢ ( 𝑀 ∈ MetSp → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ )