Metamath Proof Explorer


Theorem neicvgf1o

Description: If neighborhood and convergent functions are related by operator H , it is a one-to-one onto relation. (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 neicvgf1o ( 𝜑𝐻 : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) )

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 3 6 7 neicvgbex ( 𝜑𝐵 ∈ V )
9 8 pwexd ( 𝜑 → 𝒫 𝐵 ∈ V )
10 1 9 8 4 fsovf1od ( 𝜑𝐹 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) )
11 2 3 8 dssmapf1od ( 𝜑𝐷 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) )
12 1 8 9 5 fsovf1od ( 𝜑𝐺 : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) )
13 f1oco ( ( 𝐷 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) ∧ 𝐺 : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) ) → ( 𝐷𝐺 ) : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) )
14 11 12 13 syl2anc ( 𝜑 → ( 𝐷𝐺 ) : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) )
15 f1oco ( ( 𝐹 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) ∧ ( 𝐷𝐺 ) : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵m 𝒫 𝐵 ) ) → ( 𝐹 ∘ ( 𝐷𝐺 ) ) : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) )
16 10 14 15 syl2anc ( 𝜑 → ( 𝐹 ∘ ( 𝐷𝐺 ) ) : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) )
17 f1oeq1 ( 𝐻 = ( 𝐹 ∘ ( 𝐷𝐺 ) ) → ( 𝐻 : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) ↔ ( 𝐹 ∘ ( 𝐷𝐺 ) ) : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) ) )
18 6 17 ax-mp ( 𝐻 : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) ↔ ( 𝐹 ∘ ( 𝐷𝐺 ) ) : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) )
19 16 18 sylibr ( 𝜑𝐻 : ( 𝒫 𝒫 𝐵m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) )