Theorem cnprcl

Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Hypothesis	iscnp2.1	⊢ 𝑋 = ∪ 𝐽
Assertion	cnprcl	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝑃 ∈ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		iscnp2.1	⊢ 𝑋 = ∪ 𝐽
2		eqid	⊢ ∪ 𝐾 = ∪ 𝐾
3	1 2	iscnp2	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋 ) ∧ ( 𝐹 : 𝑋 ⟶ ∪ 𝐾 ∧ ∀ 𝑦 ∈ 𝐾 ( ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) ) ) ) )
4	3	simplbi	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋 ) )
5	4	simp3d	⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝑃 ∈ 𝑋 )