Metamath Proof Explorer


Theorem iscnrm3rlem6

Description: Lemma for iscnrm3rlem7 . (Contributed by Zhi Wang, 5-Sep-2024)

Ref Expression
Hypotheses iscnrm3rlem4.1 ( 𝜑𝐽 ∈ Top )
iscnrm3rlem4.2 ( 𝜑𝑆 𝐽 )
iscnrm3rlem5.3 ( 𝜑𝑇 𝐽 )
iscnrm3rlem6.4 ( 𝜑𝑂 ⊆ ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) )
Assertion iscnrm3rlem6 ( 𝜑 → ( 𝑂 ∈ ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ↔ 𝑂𝐽 ) )

Proof

Step Hyp Ref Expression
1 iscnrm3rlem4.1 ( 𝜑𝐽 ∈ Top )
2 iscnrm3rlem4.2 ( 𝜑𝑆 𝐽 )
3 iscnrm3rlem5.3 ( 𝜑𝑇 𝐽 )
4 iscnrm3rlem6.4 ( 𝜑𝑂 ⊆ ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) )
5 1 2 3 iscnrm3rlem5 ( 𝜑 → ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ∈ 𝐽 )
6 restopn2 ( ( 𝐽 ∈ Top ∧ ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ∈ 𝐽 ) → ( 𝑂 ∈ ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ↔ ( 𝑂𝐽𝑂 ⊆ ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ) )
7 1 5 6 syl2anc ( 𝜑 → ( 𝑂 ∈ ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ↔ ( 𝑂𝐽𝑂 ⊆ ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ) )
8 4 7 mpbiran2d ( 𝜑 → ( 𝑂 ∈ ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ↔ 𝑂𝐽 ) )