Description: The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006) (Revised by Mario Carneiro, 11-Nov-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | clscld.1 | ⊢ 𝑋 = ∪ 𝐽 | |
| Assertion | ntropn | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ∈ 𝐽 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clscld.1 | ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | 1 | ntrval | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = ∪ ( 𝐽 ∩ 𝒫 𝑆 ) ) |
| 3 | inss1 | ⊢ ( 𝐽 ∩ 𝒫 𝑆 ) ⊆ 𝐽 | |
| 4 | uniopn | ⊢ ( ( 𝐽 ∈ Top ∧ ( 𝐽 ∩ 𝒫 𝑆 ) ⊆ 𝐽 ) → ∪ ( 𝐽 ∩ 𝒫 𝑆 ) ∈ 𝐽 ) | |
| 5 | 3 4 | mpan2 | ⊢ ( 𝐽 ∈ Top → ∪ ( 𝐽 ∩ 𝒫 𝑆 ) ∈ 𝐽 ) |
| 6 | 5 | adantr | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ∪ ( 𝐽 ∩ 𝒫 𝑆 ) ∈ 𝐽 ) |
| 7 | 2 6 | eqeltrd | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ∈ 𝐽 ) |