Theorem iscnrm3lem1

Description: Lemma for iscnrm3 . Subspace topology is a topology. (Contributed by Zhi Wang, 3-Sep-2024)

	Ref	Expression
Assertion	iscnrm3lem1	⊢ ( 𝐽 ∈ Top → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ( ( 𝐽 ↾t 𝑥 ) ∈ Top ∧ 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		resttop	⊢ ( ( 𝐽 ∈ Top ∧ 𝑥 ∈ 𝐴 ) → ( 𝐽 ↾t 𝑥 ) ∈ Top )
2	1	biantrurd	⊢ ( ( 𝐽 ∈ Top ∧ 𝑥 ∈ 𝐴 ) → ( 𝜑 ↔ ( ( 𝐽 ↾t 𝑥 ) ∈ Top ∧ 𝜑 ) ) )
3	2	ralbidva	⊢ ( 𝐽 ∈ Top → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ( ( 𝐽 ↾t 𝑥 ) ∈ Top ∧ 𝜑 ) ) )