Metamath Proof Explorer


Theorem restcls2

Description: A closed set in a subspace topology is the closure in the original topology intersecting with the subspace. (Contributed by Zhi Wang, 2-Sep-2024)

Ref Expression
Hypotheses restcls2.1 ( 𝜑𝐽 ∈ Top )
restcls2.2 ( 𝜑𝑋 = 𝐽 )
restcls2.3 ( 𝜑𝑌𝑋 )
restcls2.4 ( 𝜑𝐾 = ( 𝐽t 𝑌 ) )
restcls2.5 ( 𝜑𝑆 ∈ ( Clsd ‘ 𝐾 ) )
Assertion restcls2 ( 𝜑𝑆 = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) )

Proof

Step Hyp Ref Expression
1 restcls2.1 ( 𝜑𝐽 ∈ Top )
2 restcls2.2 ( 𝜑𝑋 = 𝐽 )
3 restcls2.3 ( 𝜑𝑌𝑋 )
4 restcls2.4 ( 𝜑𝐾 = ( 𝐽t 𝑌 ) )
5 restcls2.5 ( 𝜑𝑆 ∈ ( Clsd ‘ 𝐾 ) )
6 4 fveq2d ( 𝜑 → ( cls ‘ 𝐾 ) = ( cls ‘ ( 𝐽t 𝑌 ) ) )
7 6 fveq1d ( 𝜑 → ( ( cls ‘ 𝐾 ) ‘ 𝑆 ) = ( ( cls ‘ ( 𝐽t 𝑌 ) ) ‘ 𝑆 ) )
8 cldcls ( 𝑆 ∈ ( Clsd ‘ 𝐾 ) → ( ( cls ‘ 𝐾 ) ‘ 𝑆 ) = 𝑆 )
9 5 8 syl ( 𝜑 → ( ( cls ‘ 𝐾 ) ‘ 𝑆 ) = 𝑆 )
10 3 2 sseqtrd ( 𝜑𝑌 𝐽 )
11 1 2 3 4 5 restcls2lem ( 𝜑𝑆𝑌 )
12 eqid 𝐽 = 𝐽
13 eqid ( 𝐽t 𝑌 ) = ( 𝐽t 𝑌 )
14 12 13 restcls ( ( 𝐽 ∈ Top ∧ 𝑌 𝐽𝑆𝑌 ) → ( ( cls ‘ ( 𝐽t 𝑌 ) ) ‘ 𝑆 ) = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) )
15 1 10 11 14 syl3anc ( 𝜑 → ( ( cls ‘ ( 𝐽t 𝑌 ) ) ‘ 𝑆 ) = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) )
16 7 9 15 3eqtr3d ( 𝜑𝑆 = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) )