Theorem mopni

Description: An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypothesis	mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion	mopni	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	elmopn	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ 𝐽 ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) ) )
3	2	simplbda	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) )
4		eleq1	⊢ ( 𝑦 = 𝑃 → ( 𝑦 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥 ) )
5	4	anbi1d	⊢ ( 𝑦 = 𝑃 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ↔ ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
6	5	rexbidv	⊢ ( 𝑦 = 𝑃 → ( ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
7	6	rspccv	⊢ ( ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) → ( 𝑃 ∈ 𝐴 → ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
8	3 7	syl	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → ( 𝑃 ∈ 𝐴 → ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
9	8	3impia	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑥 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) )