Theorem ntrneinex

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

	Ref	Expression
Hypotheses	ntrnei.o	⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) )
	ntrnei.f	⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
	ntrnei.r	⊢ ( 𝜑 → 𝐼 𝐹 𝑁 )
Assertion	ntrneinex	⊢ ( 𝜑 → 𝑁 ∈ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ntrnei.o	⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) )
2		ntrnei.f	⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 )
3		ntrnei.r	⊢ ( 𝜑 → 𝐼 𝐹 𝑁 )
4	1 2 3	ntrneif1o	⊢ ( 𝜑 → 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) )
5		f1orel	⊢ ( 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) → Rel 𝐹 )
6	4 5	syl	⊢ ( 𝜑 → Rel 𝐹 )
7		relelrn	⊢ ( ( Rel 𝐹 ∧ 𝐼 𝐹 𝑁 ) → 𝑁 ∈ ran 𝐹 )
8	6 3 7	syl2anc	⊢ ( 𝜑 → 𝑁 ∈ ran 𝐹 )
9		dff1o2	⊢ ( 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ↔ ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ Fun ◡ 𝐹 ∧ ran 𝐹 = ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) )
10	4 9	sylib	⊢ ( 𝜑 → ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ Fun ◡ 𝐹 ∧ ran 𝐹 = ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) )
11	10	simp3d	⊢ ( 𝜑 → ran 𝐹 = ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) )
12	8 11	eleqtrd	⊢ ( 𝜑 → 𝑁 ∈ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) )