Theorem neicvgnvor

Description: If neighborhood and convergent functions are related by operator H , the relationship holds with the functions swapped. (Contributed by RP, 11-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 )
Assertion	neicvgnvor	⊢ ( 𝜑 → 𝑀 𝐻 𝑁 )

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	1 2 3 4 5 6 7	neicvgnvo	⊢ ( 𝜑 → ◡ 𝐻 = 𝐻 )
9	8	breqd	⊢ ( 𝜑 → ( 𝑁 ◡ 𝐻 𝑀 ↔ 𝑁 𝐻 𝑀 ) )
10	7 9	mpbird	⊢ ( 𝜑 → 𝑁 ◡ 𝐻 𝑀 )
11		relco	⊢ Rel ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) )
12	6	releqi	⊢ ( Rel 𝐻 ↔ Rel ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) )
13	11 12	mpbir	⊢ Rel 𝐻
14	13	relbrcnv	⊢ ( 𝑁 ◡ 𝐻 𝑀 ↔ 𝑀 𝐻 𝑁 )
15	10 14	sylib	⊢ ( 𝜑 → 𝑀 𝐻 𝑁 )