Description: Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015) (Revised by Mario Carneiro, 1-May-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | restrcl | ⊢ ( ( 𝐽 ↾t 𝐴 ) ∈ Top → ( 𝐽 ∈ V ∧ 𝐴 ∈ V ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0opn | ⊢ ( ( 𝐽 ↾t 𝐴 ) ∈ Top → ∅ ∈ ( 𝐽 ↾t 𝐴 ) ) | |
2 | n0i | ⊢ ( ∅ ∈ ( 𝐽 ↾t 𝐴 ) → ¬ ( 𝐽 ↾t 𝐴 ) = ∅ ) | |
3 | 1 2 | syl | ⊢ ( ( 𝐽 ↾t 𝐴 ) ∈ Top → ¬ ( 𝐽 ↾t 𝐴 ) = ∅ ) |
4 | restfn | ⊢ ↾t Fn ( V × V ) | |
5 | 4 | fndmi | ⊢ dom ↾t = ( V × V ) |
6 | 5 | ndmov | ⊢ ( ¬ ( 𝐽 ∈ V ∧ 𝐴 ∈ V ) → ( 𝐽 ↾t 𝐴 ) = ∅ ) |
7 | 3 6 | nsyl2 | ⊢ ( ( 𝐽 ↾t 𝐴 ) ∈ Top → ( 𝐽 ∈ V ∧ 𝐴 ∈ V ) ) |