Metamath Proof Explorer


Theorem iscnrm3lem1

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

Ref Expression
Assertion iscnrm3lem1 ( 𝐽 ∈ Top → ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑥𝐴 ( ( 𝐽t 𝑥 ) ∈ Top ∧ 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 resttop ( ( 𝐽 ∈ Top ∧ 𝑥𝐴 ) → ( 𝐽t 𝑥 ) ∈ Top )
2 1 biantrurd ( ( 𝐽 ∈ Top ∧ 𝑥𝐴 ) → ( 𝜑 ↔ ( ( 𝐽t 𝑥 ) ∈ Top ∧ 𝜑 ) ) )
3 2 ralbidva ( 𝐽 ∈ Top → ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑥𝐴 ( ( 𝐽t 𝑥 ) ∈ Top ∧ 𝜑 ) ) )