Theorem ntrneicls00

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F , then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-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	ntrneicls00	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝐵 ) = 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝐵 ∈ ( 𝑁 ‘ 𝑥 ) ) )

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	ntrneiiex	⊢ ( 𝜑 → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
5		elmapi	⊢ ( 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
6	4 5	syl	⊢ ( 𝜑 → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
7	1 2 3	ntrneibex	⊢ ( 𝜑 → 𝐵 ∈ V )
8		pwidg	⊢ ( 𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵 )
9	7 8	syl	⊢ ( 𝜑 → 𝐵 ∈ 𝒫 𝐵 )
10	6 9	ffvelrnd	⊢ ( 𝜑 → ( 𝐼 ‘ 𝐵 ) ∈ 𝒫 𝐵 )
11	10	elpwid	⊢ ( 𝜑 → ( 𝐼 ‘ 𝐵 ) ⊆ 𝐵 )
12		eqss	⊢ ( ( 𝐼 ‘ 𝐵 ) = 𝐵 ↔ ( ( 𝐼 ‘ 𝐵 ) ⊆ 𝐵 ∧ 𝐵 ⊆ ( 𝐼 ‘ 𝐵 ) ) )
13		dfss3	⊢ ( 𝐵 ⊆ ( 𝐼 ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ ( 𝐼 ‘ 𝐵 ) )
14	13	anbi2i	⊢ ( ( ( 𝐼 ‘ 𝐵 ) ⊆ 𝐵 ∧ 𝐵 ⊆ ( 𝐼 ‘ 𝐵 ) ) ↔ ( ( 𝐼 ‘ 𝐵 ) ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ ( 𝐼 ‘ 𝐵 ) ) )
15	12 14	bitri	⊢ ( ( 𝐼 ‘ 𝐵 ) = 𝐵 ↔ ( ( 𝐼 ‘ 𝐵 ) ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ ( 𝐼 ‘ 𝐵 ) ) )
16	15	a1i	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝐵 ) = 𝐵 ↔ ( ( 𝐼 ‘ 𝐵 ) ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ ( 𝐼 ‘ 𝐵 ) ) ) )
17	11 16	mpbirand	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝐵 ) = 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ ( 𝐼 ‘ 𝐵 ) ) )
18	3	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐼 𝐹 𝑁 )
19		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 )
20	9	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐵 ∈ 𝒫 𝐵 )
21	1 2 18 19 20	ntrneiel	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ∈ ( 𝐼 ‘ 𝐵 ) ↔ 𝐵 ∈ ( 𝑁 ‘ 𝑥 ) ) )
22	21	ralbidva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝑥 ∈ ( 𝐼 ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐵 𝐵 ∈ ( 𝑁 ‘ 𝑥 ) ) )
23	17 22	bitrd	⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝐵 ) = 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝐵 ∈ ( 𝑁 ‘ 𝑥 ) ) )