Theorem mopni2

Description: An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypothesis	mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion	mopni2	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mopni.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	mopni	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) )
3	1	mopnss	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝑋 )
4	3	sselda	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ 𝑋 )
5		blssex	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ( ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 ) )
6	5	adantlr	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) ∧ 𝑃 ∈ 𝑋 ) → ( ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 ) )
7	4 6	syldan	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ) ∧ 𝑃 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 ) )
8	7	3impa	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ( ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑃 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 ) )
9	2 8	mpbid	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℝ+ ( 𝑃 ( ball ‘ 𝐷 ) 𝑥 ) ⊆ 𝐴 )