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Metamath Proof Explorer

Theorem elmopn

Description: The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006) (Revised by Mario Carneiro, 12-Nov-2013)

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

Proof

Step Hyp Ref Expression
1 mopnval.1 ⊢ 𝐽 = ( MetOpen ‘ 𝐷 )
2 1 mopnval ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 = ( topGen ‘ ran ( ball ‘ 𝐷 ) ) )
3 2 eleq2d ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ 𝐽 ↔ 𝐴 ∈ ( topGen ‘ ran ( ball ‘ 𝐷 ) ) ) )
4 blbas ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ran ( ball ‘ 𝐷 ) ∈ TopBases )
5 eltg2 ⊢ ( ran ( ball ‘ 𝐷 ) ∈ TopBases → ( 𝐴 ∈ ( topGen ‘ ran ( ball ‘ 𝐷 ) ) ↔ ( 𝐴 ⊆ ∪ ran ( ball ‘ 𝐷 ) ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )
6 4 5 syl ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ ( topGen ‘ ran ( ball ‘ 𝐷 ) ) ↔ ( 𝐴 ⊆ ∪ ran ( ball ‘ 𝐷 ) ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )
7 unirnbl ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∪ ran ( ball ‘ 𝐷 ) = 𝑋 )
8 7 sseq2d ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ⊆ ∪ ran ( ball ‘ 𝐷 ) ↔ 𝐴 ⊆ 𝑋 ) )
9 8 anbi1d ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ( 𝐴 ⊆ ∪ ran ( ball ‘ 𝐷 ) ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )
10 3 6 9 3bitrd ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐴 ∈ 𝐽 ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ ran ( ball ‘ 𝐷 ) ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )