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Metamath Proof Explorer

Theorem cnf

Description: A continuous function is a mapping. (Contributed by FL, 8-Dec-2006) (Revised by Mario Carneiro, 21-Aug-2015)

Ref Expression
Hypotheses iscnp2.1 ⊢ 𝑋 = ∪ 𝐽
iscnp2.2 ⊢ 𝑌 = ∪ 𝐾
Assertion cnf ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐹 : 𝑋 ⟶ 𝑌 )

Proof

Step Hyp Ref Expression
1 iscnp2.1 ⊢ 𝑋 = ∪ 𝐽
2 iscnp2.2 ⊢ 𝑌 = ∪ 𝐾
3 1 2 iscn2 ⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) ∧ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) )
4 3 simprbi ⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) )
5 4 simpld ⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐹 : 𝑋 ⟶ 𝑌 )