Theorem clsneifv3

Description: Value of the neighborhoods (convergents) in terms of the closure (interior) function. (Contributed by RP, 27-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 )
	clsneifv.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	clsneifv3	⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) = { 𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ ( 𝐾 ‘ ( 𝐵 ∖ 𝑠 ) ) } )

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		clsneifv.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8		dfin5	⊢ ( 𝒫 𝐵 ∩ ( 𝑁 ‘ 𝑋 ) ) = { 𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) }
9	1 2 3 4 5 6	clsneinex	⊢ ( 𝜑 → 𝑁 ∈ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) )
10		elmapi	⊢ ( 𝑁 ∈ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) → 𝑁 : 𝐵 ⟶ 𝒫 𝒫 𝐵 )
11	9 10	syl	⊢ ( 𝜑 → 𝑁 : 𝐵 ⟶ 𝒫 𝒫 𝐵 )
12	11 7	ffvelcdmd	⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) ∈ 𝒫 𝒫 𝐵 )
13	12	elpwid	⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) ⊆ 𝒫 𝐵 )
14		sseqin2	⊢ ( ( 𝑁 ‘ 𝑋 ) ⊆ 𝒫 𝐵 ↔ ( 𝒫 𝐵 ∩ ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ 𝑋 ) )
15	13 14	sylib	⊢ ( 𝜑 → ( 𝒫 𝐵 ∩ ( 𝑁 ‘ 𝑋 ) ) = ( 𝑁 ‘ 𝑋 ) )
16	6	adantr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝐾 𝐻 𝑁 )
17	7	adantr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝑋 ∈ 𝐵 )
18		simpr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝑠 ∈ 𝒫 𝐵 )
19	1 2 3 4 5 16 17 18	clsneiel2	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → ( 𝑋 ∈ ( 𝐾 ‘ ( 𝐵 ∖ 𝑠 ) ) ↔ ¬ 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) )
20	19	con2bid	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → ( 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ↔ ¬ 𝑋 ∈ ( 𝐾 ‘ ( 𝐵 ∖ 𝑠 ) ) ) )
21	20	rabbidva	⊢ ( 𝜑 → { 𝑠 ∈ 𝒫 𝐵 ∣ 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) } = { 𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ ( 𝐾 ‘ ( 𝐵 ∖ 𝑠 ) ) } )
22	8 15 21	3eqtr3a	⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) = { 𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ ( 𝐾 ‘ ( 𝐵 ∖ 𝑠 ) ) } )