Metamath Proof Explorer


Theorem ntrclsiex

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 𝑖m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖𝑗 ) ) ) ) ) )
ntrcls.d 𝐷 = ( 𝑂𝐵 )
ntrcls.r ( 𝜑𝐼 𝐷 𝐾 )
Assertion ntrclsiex ( 𝜑𝐼 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ntrcls.o 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖𝑗 ) ) ) ) ) )
2 ntrcls.d 𝐷 = ( 𝑂𝐵 )
3 ntrcls.r ( 𝜑𝐼 𝐷 𝐾 )
4 1 2 3 ntrclsf1o ( 𝜑𝐷 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) )
5 f1orel ( 𝐷 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) → Rel 𝐷 )
6 4 5 syl ( 𝜑 → Rel 𝐷 )
7 releldm ( ( Rel 𝐷𝐼 𝐷 𝐾 ) → 𝐼 ∈ dom 𝐷 )
8 6 3 7 syl2anc ( 𝜑𝐼 ∈ dom 𝐷 )
9 f1odm ( 𝐷 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) → dom 𝐷 = ( 𝒫 𝐵m 𝒫 𝐵 ) )
10 4 9 syl ( 𝜑 → dom 𝐷 = ( 𝒫 𝐵m 𝒫 𝐵 ) )
11 8 10 eleqtrd ( 𝜑𝐼 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) )