Description: An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | isopn3i | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → 𝑆 ∈ 𝐽 ) | |
2 | elssuni | ⊢ ( 𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ 𝐽 ) | |
3 | eqid | ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ) → ( 𝑆 ∈ 𝐽 ↔ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 ) ) |
5 | 2 4 | sylan2 | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → ( 𝑆 ∈ 𝐽 ↔ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 ) ) |
6 | 1 5 | mpbid | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 ) |