Metamath Proof Explorer


Theorem iscnrm3rlem7

Description: Lemma for iscnrm3rlem8 . Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024)

Ref Expression
Hypotheses iscnrm3rlem4.1 ( 𝜑𝐽 ∈ Top )
iscnrm3rlem4.2 ( 𝜑𝑆 𝐽 )
iscnrm3rlem5.3 ( 𝜑𝑇 𝐽 )
iscnrm3rlem7.4 ( 𝜑𝑂 ∈ ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) )
Assertion iscnrm3rlem7 ( 𝜑𝑂𝐽 )

Proof

Step Hyp Ref Expression
1 iscnrm3rlem4.1 ( 𝜑𝐽 ∈ Top )
2 iscnrm3rlem4.2 ( 𝜑𝑆 𝐽 )
3 iscnrm3rlem5.3 ( 𝜑𝑇 𝐽 )
4 iscnrm3rlem7.4 ( 𝜑𝑂 ∈ ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) )
5 1 uniexd ( 𝜑 𝐽 ∈ V )
6 5 difexd ( 𝜑 → ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ∈ V )
7 resttop ( ( 𝐽 ∈ Top ∧ ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ∈ V ) → ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ∈ Top )
8 1 6 7 syl2anc ( 𝜑 → ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ∈ Top )
9 eqid ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) = ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) )
10 9 eltopss ( ( ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ∈ Top ∧ 𝑂 ∈ ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ) → 𝑂 ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) )
11 8 4 10 syl2anc ( 𝜑𝑂 ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) )
12 difssd ( 𝜑 → ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ⊆ 𝐽 )
13 eqid 𝐽 = 𝐽
14 13 restuni ( ( 𝐽 ∈ Top ∧ ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ⊆ 𝐽 ) → ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) = ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) )
15 1 12 14 syl2anc ( 𝜑 → ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) = ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) )
16 11 15 sseqtrrd ( 𝜑𝑂 ⊆ ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) )
17 1 2 3 16 iscnrm3rlem6 ( 𝜑 → ( 𝑂 ∈ ( 𝐽t ( 𝐽 ∖ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑇 ) ) ) ) ↔ 𝑂𝐽 ) )
18 4 17 mpbid ( 𝜑𝑂𝐽 )