Theorem neicvgel2

Description: The complement of a subset being an element of a neighborhood at a point is equivalent to that subset not being a element of the convergent at that point. (Contributed by RP, 12-Jun-2021)

	Ref	Expression
Hypotheses	neicvg.o	⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) )
	neicvg.p	⊢ 𝑃 = ( 𝑛 ∈ V ↦ ( 𝑝 ∈ ( 𝒫 𝑛 ↑m 𝒫 𝑛 ) ↦ ( 𝑜 ∈ 𝒫 𝑛 ↦ ( 𝑛 ∖ ( 𝑝 ‘ ( 𝑛 ∖ 𝑜 ) ) ) ) ) )
	neicvg.d	⊢ 𝐷 = ( 𝑃 ‘ 𝐵 )
	neicvg.f	⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
	neicvg.g	⊢ 𝐺 = ( 𝐵 𝑂 𝒫 𝐵 )
	neicvg.h	⊢ 𝐻 = ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) )
	neicvg.r	⊢ ( 𝜑 → 𝑁 𝐻 𝑀 )
	neicvgel.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	neicvgel.s	⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 )
Assertion	neicvgel2	⊢ ( 𝜑 → ( ( 𝐵 ∖ 𝑆 ) ∈ ( 𝑁 ‘ 𝑋 ) ↔ ¬ 𝑆 ∈ ( 𝑀 ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		neicvg.o	⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) )
2		neicvg.p	⊢ 𝑃 = ( 𝑛 ∈ V ↦ ( 𝑝 ∈ ( 𝒫 𝑛 ↑m 𝒫 𝑛 ) ↦ ( 𝑜 ∈ 𝒫 𝑛 ↦ ( 𝑛 ∖ ( 𝑝 ‘ ( 𝑛 ∖ 𝑜 ) ) ) ) ) )
3		neicvg.d	⊢ 𝐷 = ( 𝑃 ‘ 𝐵 )
4		neicvg.f	⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
5		neicvg.g	⊢ 𝐺 = ( 𝐵 𝑂 𝒫 𝐵 )
6		neicvg.h	⊢ 𝐻 = ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) )
7		neicvg.r	⊢ ( 𝜑 → 𝑁 𝐻 𝑀 )
8		neicvgel.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
9		neicvgel.s	⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 )
10	3 6 7	neicvgrcomplex	⊢ ( 𝜑 → ( 𝐵 ∖ 𝑆 ) ∈ 𝒫 𝐵 )
11	1 2 3 4 5 6 7 8 10	neicvgel1	⊢ ( 𝜑 → ( ( 𝐵 ∖ 𝑆 ) ∈ ( 𝑁 ‘ 𝑋 ) ↔ ¬ ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) ∈ ( 𝑀 ‘ 𝑋 ) ) )
12	9	elpwid	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
13		dfss4	⊢ ( 𝑆 ⊆ 𝐵 ↔ ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) = 𝑆 )
14	12 13	sylib	⊢ ( 𝜑 → ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) = 𝑆 )
15	14	eleq1d	⊢ ( 𝜑 → ( ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) ∈ ( 𝑀 ‘ 𝑋 ) ↔ 𝑆 ∈ ( 𝑀 ‘ 𝑋 ) ) )
16	15	notbid	⊢ ( 𝜑 → ( ¬ ( 𝐵 ∖ ( 𝐵 ∖ 𝑆 ) ) ∈ ( 𝑀 ‘ 𝑋 ) ↔ ¬ 𝑆 ∈ ( 𝑀 ‘ 𝑋 ) ) )
17	11 16	bitrd	⊢ ( 𝜑 → ( ( 𝐵 ∖ 𝑆 ) ∈ ( 𝑁 ‘ 𝑋 ) ↔ ¬ 𝑆 ∈ ( 𝑀 ‘ 𝑋 ) ) )