Theorem cnima

Description: An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006)

	Ref	Expression
Assertion	cnima	⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝐾 ) → ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 )

Proof

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	simprbi	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → ( 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ∧ ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) )
5	4	simprd	⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 )
6		imaeq2	⊢ ( 𝑥 = 𝐴 → ( ◡ 𝐹 “ 𝑥 ) = ( ◡ 𝐹 “ 𝐴 ) )
7	6	eleq1d	⊢ ( 𝑥 = 𝐴 → ( ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ↔ ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 ) )
8	7	rspccva	⊢ ( ( ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ∧ 𝐴 ∈ 𝐾 ) → ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 )
9	5 8	sylan	⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ 𝐴 ∈ 𝐾 ) → ( ◡ 𝐹 “ 𝐴 ) ∈ 𝐽 )