Metamath Proof Explorer


Theorem ntrclsfv2

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-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 ntrclsfv2 ( 𝜑 → ( 𝐷𝐾 ) = 𝐼 )

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 ntrclsnvobr ( 𝜑𝐾 𝐷 𝐼 )
5 1 2 4 ntrclsfv1 ( 𝜑 → ( 𝐷𝐾 ) = 𝐼 )