Description: Reverse closure for a continuous function. (Contributed by Mario Carneiro, 21-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | cntop1 | ⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐽 ∈ Top ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid | ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | eqid | ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 1 2 | iscn2 | ⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) ∧ ( 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ∧ ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
4 | 3 | simplbi | ⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) ) |
5 | 4 | simpld | ⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐽 ∈ Top ) |