Theorem ntrclsf1o

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator we may characterize the relation as part of a 1-to-1 onto function. (Contributed by RP, 29-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	ntrclsf1o	⊢ ( 𝜑 → 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 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	2 3	ntrclsbex	⊢ ( 𝜑 → 𝐵 ∈ V )
5	1 2 4	dssmapf1od	⊢ ( 𝜑 → 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )