Theorem elmopn2

Description: A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007) (Revised by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypothesis	mopnval.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion	elmopn2	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ 𝐽 ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mopnval.1	⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2	1	elmopn	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ 𝐽 ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴 ) ) ) )
3		ssel2	⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝑋 )
4		blssex	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ) → ( ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴 ) ↔ ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) )
5	3 4	sylan2	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐴 ) ) → ( ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴 ) ↔ ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) )
6	5	anassrs	⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) ∧ 𝑥 ∈ 𝐴 ) → ( ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴 ) ↔ ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) )
7	6	ralbidva	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴 ) ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) )
8	7	pm5.32da	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝐴 ) ) ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) ) )
9	2 8	bitrd	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ 𝐽 ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑦 ) ⊆ 𝐴 ) ) )