Description: If A is open, then A is open in the restriction to itself. (Contributed by Glauco Siliprandi, 21-Dec-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | restopn3 | ⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ∈ ( 𝐽 ↾t 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr | ⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ∈ 𝐽 ) | |
2 | ssidd | ⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝐴 ) | |
3 | restopn2 | ⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ) → ( 𝐴 ∈ ( 𝐽 ↾t 𝐴 ) ↔ ( 𝐴 ∈ 𝐽 ∧ 𝐴 ⊆ 𝐴 ) ) ) | |
4 | 1 2 3 | mpbir2and | ⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ∈ ( 𝐽 ↾t 𝐴 ) ) |