Theorem unirnbl

Description: The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Assertion	unirnbl	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∪ ran ( ball ‘ 𝐷 ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		blf	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ball ‘ 𝐷 ) : ( 𝑋 × ℝ* ) ⟶ 𝒫 𝑋 )
2	1	frnd	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) ⊆ 𝒫 𝑋 )
3		sspwuni	⊢ ( ran ( ball ‘ 𝐷 ) ⊆ 𝒫 𝑋 ↔ ∪ ran ( ball ‘ 𝐷 ) ⊆ 𝑋 )
4	2 3	sylib	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∪ ran ( ball ‘ 𝐷 ) ⊆ 𝑋 )
5		1rp	⊢ 1 ∈ ℝ+
6		blcntr	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ+ ) → 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) 1 ) )
7	5 6	mp3an3	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) 1 ) )
8		1xr	⊢ 1 ∈ ℝ*
9		blelrn	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 1 ∈ ℝ* ) → ( 𝑥 ( ball ‘ 𝐷 ) 1 ) ∈ ran ( ball ‘ 𝐷 ) )
10	8 9	mp3an3	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝑥 ( ball ‘ 𝐷 ) 1 ) ∈ ran ( ball ‘ 𝐷 ) )
11		elunii	⊢ ( ( 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) 1 ) ∧ ( 𝑥 ( ball ‘ 𝐷 ) 1 ) ∈ ran ( ball ‘ 𝐷 ) ) → 𝑥 ∈ ∪ ran ( ball ‘ 𝐷 ) )
12	7 10 11	syl2anc	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → 𝑥 ∈ ∪ ran ( ball ‘ 𝐷 ) )
13	4 12	eqelssd	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∪ ran ( ball ‘ 𝐷 ) = 𝑋 )