Theorem ntrneicnv

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F , then converse of F is known. (Contributed by RP, 29-May-2021)

	Ref	Expression
Hypotheses	ntrnei.o	⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) )
	ntrnei.f	⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
	ntrnei.r	⊢ ( 𝜑 → 𝐼 𝐹 𝑁 )
Assertion	ntrneicnv	⊢ ( 𝜑 → ◡ 𝐹 = ( 𝐵 𝑂 𝒫 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ntrnei.o	⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) )
2		ntrnei.f	⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
3		ntrnei.r	⊢ ( 𝜑 → 𝐼 𝐹 𝑁 )
4	1 2 3	ntrneibex	⊢ ( 𝜑 → 𝐵 ∈ V )
5	4	pwexd	⊢ ( 𝜑 → 𝒫 𝐵 ∈ V )
6		eqid	⊢ ( 𝐵 𝑂 𝒫 𝐵 ) = ( 𝐵 𝑂 𝒫 𝐵 )
7	1 5 4 2 6	fsovcnvd	⊢ ( 𝜑 → ◡ 𝐹 = ( 𝐵 𝑂 𝒫 𝐵 ) )