Metamath Proof Explorer


Theorem neicvgnvo

Description: If neighborhood and convergent functions are related by operator H , it is its own converse function. (Contributed by RP, 11-Jun-2021)

Ref Expression
Hypotheses neicvg.o 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗m 𝑖 ) ↦ ( 𝑙𝑗 ↦ { 𝑚𝑖𝑙 ∈ ( 𝑘𝑚 ) } ) ) )
neicvg.p 𝑃 = ( 𝑛 ∈ V ↦ ( 𝑝 ∈ ( 𝒫 𝑛m 𝒫 𝑛 ) ↦ ( 𝑜 ∈ 𝒫 𝑛 ↦ ( 𝑛 ∖ ( 𝑝 ‘ ( 𝑛𝑜 ) ) ) ) ) )
neicvg.d 𝐷 = ( 𝑃𝐵 )
neicvg.f 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
neicvg.g 𝐺 = ( 𝐵 𝑂 𝒫 𝐵 )
neicvg.h 𝐻 = ( 𝐹 ∘ ( 𝐷𝐺 ) )
neicvg.r ( 𝜑𝑁 𝐻 𝑀 )
Assertion neicvgnvo ( 𝜑 𝐻 = 𝐻 )

Proof

Step Hyp Ref Expression
1 neicvg.o 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗m 𝑖 ) ↦ ( 𝑙𝑗 ↦ { 𝑚𝑖𝑙 ∈ ( 𝑘𝑚 ) } ) ) )
2 neicvg.p 𝑃 = ( 𝑛 ∈ V ↦ ( 𝑝 ∈ ( 𝒫 𝑛m 𝒫 𝑛 ) ↦ ( 𝑜 ∈ 𝒫 𝑛 ↦ ( 𝑛 ∖ ( 𝑝 ‘ ( 𝑛𝑜 ) ) ) ) ) )
3 neicvg.d 𝐷 = ( 𝑃𝐵 )
4 neicvg.f 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
5 neicvg.g 𝐺 = ( 𝐵 𝑂 𝒫 𝐵 )
6 neicvg.h 𝐻 = ( 𝐹 ∘ ( 𝐷𝐺 ) )
7 neicvg.r ( 𝜑𝑁 𝐻 𝑀 )
8 6 cnveqi 𝐻 = ( 𝐹 ∘ ( 𝐷𝐺 ) )
9 cnvco ( 𝐹 ∘ ( 𝐷𝐺 ) ) = ( ( 𝐷𝐺 ) ∘ 𝐹 )
10 cnvco ( 𝐷𝐺 ) = ( 𝐺 𝐷 )
11 10 coeq1i ( ( 𝐷𝐺 ) ∘ 𝐹 ) = ( ( 𝐺 𝐷 ) ∘ 𝐹 )
12 8 9 11 3eqtri 𝐻 = ( ( 𝐺 𝐷 ) ∘ 𝐹 )
13 3 6 7 neicvgbex ( 𝜑𝐵 ∈ V )
14 13 pwexd ( 𝜑 → 𝒫 𝐵 ∈ V )
15 1 13 14 5 4 fsovcnvd ( 𝜑 𝐺 = 𝐹 )
16 2 3 13 dssmapnvod ( 𝜑 𝐷 = 𝐷 )
17 15 16 coeq12d ( 𝜑 → ( 𝐺 𝐷 ) = ( 𝐹𝐷 ) )
18 1 14 13 4 5 fsovcnvd ( 𝜑 𝐹 = 𝐺 )
19 17 18 coeq12d ( 𝜑 → ( ( 𝐺 𝐷 ) ∘ 𝐹 ) = ( ( 𝐹𝐷 ) ∘ 𝐺 ) )
20 12 19 syl5eq ( 𝜑 𝐻 = ( ( 𝐹𝐷 ) ∘ 𝐺 ) )
21 coass ( ( 𝐹𝐷 ) ∘ 𝐺 ) = ( 𝐹 ∘ ( 𝐷𝐺 ) )
22 21 6 eqtr4i ( ( 𝐹𝐷 ) ∘ 𝐺 ) = 𝐻
23 20 22 eqtrdi ( 𝜑 𝐻 = 𝐻 )