Theorem clsneiel2

Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the H operator, then membership in the closure of the complement of a subset is equivalent to the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021)

	Ref	Expression
Hypotheses	clsnei.o	⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) )
	clsnei.p	⊢ 𝑃 = ( 𝑛 ∈ V ↦ ( 𝑝 ∈ ( 𝒫 𝑛 ↑m 𝒫 𝑛 ) ↦ ( 𝑜 ∈ 𝒫 𝑛 ↦ ( 𝑛 ∖ ( 𝑝 ‘ ( 𝑛 ∖ 𝑜 ) ) ) ) ) )
	clsnei.d	⊢ 𝐷 = ( 𝑃 ‘ 𝐵 )
	clsnei.f	⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
	clsnei.h	⊢ 𝐻 = ( 𝐹 ∘ 𝐷 )
	clsnei.r	⊢ ( 𝜑 → 𝐾 𝐻 𝑁 )
	clsneiel.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	clsneiel.s	⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 )
Assertion	clsneiel2	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐾 ‘ ( 𝐵 ∖ 𝑆 ) ) ↔ ¬ 𝑆 ∈ ( 𝑁 ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		clsnei.o	⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) )
2		clsnei.p	⊢ 𝑃 = ( 𝑛 ∈ V ↦ ( 𝑝 ∈ ( 𝒫 𝑛 ↑m 𝒫 𝑛 ) ↦ ( 𝑜 ∈ 𝒫 𝑛 ↦ ( 𝑛 ∖ ( 𝑝 ‘ ( 𝑛 ∖ 𝑜 ) ) ) ) ) )
3		clsnei.d	⊢ 𝐷 = ( 𝑃 ‘ 𝐵 )
4		clsnei.f	⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
5		clsnei.h	⊢ 𝐻 = ( 𝐹 ∘ 𝐷 )
6		clsnei.r	⊢ ( 𝜑 → 𝐾 𝐻 𝑁 )
7		clsneiel.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8		clsneiel.s	⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 )
9	3 5 6	clsneircomplex	⊢ ( 𝜑 → ( 𝐵 ∖ 𝑆 ) ∈ 𝒫 𝐵 )
10	1 2 3 4 5 6 7 9	clsneiel1	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐾 ‘ ( 𝐵 ∖ 𝑆 ) ) ↔ ¬ ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) ∈ ( 𝑁 ‘ 𝑋 ) ) )
11	8	elpwid	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
12		dfss4	⊢ ( 𝑆 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) = 𝑆 )
13	11 12	sylib	⊢ ( 𝜑 → ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) = 𝑆 )
14	13	eleq1d	⊢ ( 𝜑 → ( ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) ∈ ( 𝑁 ‘ 𝑋 ) ↔ 𝑆 ∈ ( 𝑁 ‘ 𝑋 ) ) )
15	14	notbid	⊢ ( 𝜑 → ( ¬ ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) ∈ ( 𝑁 ‘ 𝑋 ) ↔ ¬ 𝑆 ∈ ( 𝑁 ‘ 𝑋 ) ) )
16	10 15	bitrd	⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝐾 ‘ ( 𝐵 ∖ 𝑆 ) ) ↔ ¬ 𝑆 ∈ ( 𝑁 ‘ 𝑋 ) ) )