Metamath Proof Explorer


Theorem ntrneifv1

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F , then the function value of F is the neighborhood function. (Contributed by RP, 29-May-2021)

Ref Expression
Hypotheses ntrnei.o 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗m 𝑖 ) ↦ ( 𝑙𝑗 ↦ { 𝑚𝑖𝑙 ∈ ( 𝑘𝑚 ) } ) ) )
ntrnei.f 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
ntrnei.r ( 𝜑𝐼 𝐹 𝑁 )
Assertion ntrneifv1 ( 𝜑 → ( 𝐹𝐼 ) = 𝑁 )

Proof

Step Hyp Ref Expression
1 ntrnei.o 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗m 𝑖 ) ↦ ( 𝑙𝑗 ↦ { 𝑚𝑖𝑙 ∈ ( 𝑘𝑚 ) } ) ) )
2 ntrnei.f 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
3 ntrnei.r ( 𝜑𝐼 𝐹 𝑁 )
4 1 2 3 ntrneif1o ( 𝜑𝐹 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) )
5 f1ofn ( 𝐹 : ( 𝒫 𝐵m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵m 𝐵 ) → 𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) )
6 4 5 syl ( 𝜑𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) )
7 1 2 3 ntrneiiex ( 𝜑𝐼 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) )
8 6 7 jca ( 𝜑 → ( 𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) ∧ 𝐼 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) ) )
9 fnfun ( 𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) → Fun 𝐹 )
10 9 adantr ( ( 𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) ∧ 𝐼 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) ) → Fun 𝐹 )
11 fndm ( 𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) → dom 𝐹 = ( 𝒫 𝐵m 𝒫 𝐵 ) )
12 11 eleq2d ( 𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) → ( 𝐼 ∈ dom 𝐹𝐼 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) ) )
13 12 biimpar ( ( 𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) ∧ 𝐼 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) ) → 𝐼 ∈ dom 𝐹 )
14 10 13 jca ( ( 𝐹 Fn ( 𝒫 𝐵m 𝒫 𝐵 ) ∧ 𝐼 ∈ ( 𝒫 𝐵m 𝒫 𝐵 ) ) → ( Fun 𝐹𝐼 ∈ dom 𝐹 ) )
15 funbrfvb ( ( Fun 𝐹𝐼 ∈ dom 𝐹 ) → ( ( 𝐹𝐼 ) = 𝑁𝐼 𝐹 𝑁 ) )
16 8 14 15 3syl ( 𝜑 → ( ( 𝐹𝐼 ) = 𝑁𝐼 𝐹 𝑁 ) )
17 3 16 mpbird ( 𝜑 → ( 𝐹𝐼 ) = 𝑁 )