Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Richard Penner
Exploring Topology via Seifert and Threlfall
Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods
ntrclskex
Metamath Proof Explorer
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the
duality operator then those functions are maps of subsets to subsets.
(Contributed by RP , 21-May-2021)
Ref
Expression
Hypotheses
ntrcls.o
⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) )
ntrcls.d
⊢ 𝐷 = ( 𝑂 ‘ 𝐵 )
ntrcls.r
⊢ ( 𝜑 → 𝐼 𝐷 𝐾 )
Assertion
ntrclskex
⊢ ( 𝜑 → 𝐾 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
Proof
Step
Hyp
Ref
Expression
1
ntrcls.o
⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) )
2
ntrcls.d
⊢ 𝐷 = ( 𝑂 ‘ 𝐵 )
3
ntrcls.r
⊢ ( 𝜑 → 𝐼 𝐷 𝐾 )
4
1 2 3
ntrclsnvobr
⊢ ( 𝜑 → 𝐾 𝐷 𝐼 )
5
1 2 4
ntrclsiex
⊢ ( 𝜑 → 𝐾 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )