Description: A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | cnf2 | ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscn | ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) | |
2 | 1 | simprbda | ⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |
3 | 2 | 3impa | ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |