Metamath Proof Explorer


Theorem gneispacef2

Description: A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021)

Ref Expression
Hypothesis gneispace.a 𝐴 = { 𝑓 ∣ ( 𝑓 : dom 𝑓 ⟶ ( 𝒫 ( 𝒫 dom 𝑓 ∖ { ∅ } ) ∖ { ∅ } ) ∧ ∀ 𝑝 ∈ dom 𝑓𝑛 ∈ ( 𝑓𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝑓 ( 𝑛𝑠𝑠 ∈ ( 𝑓𝑝 ) ) ) ) }
Assertion gneispacef2 ( 𝐹𝐴𝐹 : dom 𝐹 ⟶ 𝒫 𝒫 dom 𝐹 )

Proof

Step Hyp Ref Expression
1 gneispace.a 𝐴 = { 𝑓 ∣ ( 𝑓 : dom 𝑓 ⟶ ( 𝒫 ( 𝒫 dom 𝑓 ∖ { ∅ } ) ∖ { ∅ } ) ∧ ∀ 𝑝 ∈ dom 𝑓𝑛 ∈ ( 𝑓𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝑓 ( 𝑛𝑠𝑠 ∈ ( 𝑓𝑝 ) ) ) ) }
2 elex ( 𝐹𝐴𝐹 ∈ V )
3 1 gneispace ( 𝐹 ∈ V → ( 𝐹𝐴 ↔ ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛𝑠𝑠 ∈ ( 𝐹𝑝 ) ) ) ) ) ) )
4 2 3 syl ( 𝐹𝐴 → ( 𝐹𝐴 ↔ ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛𝑠𝑠 ∈ ( 𝐹𝑝 ) ) ) ) ) ) )
5 4 ibi ( 𝐹𝐴 → ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛𝑠𝑠 ∈ ( 𝐹𝑝 ) ) ) ) ) )
6 simp1 ( ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛𝑠𝑠 ∈ ( 𝐹𝑝 ) ) ) ) ) → Fun 𝐹 )
7 6 funfnd ( ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛𝑠𝑠 ∈ ( 𝐹𝑝 ) ) ) ) ) → 𝐹 Fn dom 𝐹 )
8 simp2 ( ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛𝑠𝑠 ∈ ( 𝐹𝑝 ) ) ) ) ) → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 )
9 df-f ( 𝐹 : dom 𝐹 ⟶ 𝒫 𝒫 dom 𝐹 ↔ ( 𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ) )
10 7 8 9 sylanbrc ( ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹𝑝 ) ( 𝑝𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛𝑠𝑠 ∈ ( 𝐹𝑝 ) ) ) ) ) → 𝐹 : dom 𝐹 ⟶ 𝒫 𝒫 dom 𝐹 )
11 5 10 syl ( 𝐹𝐴𝐹 : dom 𝐹 ⟶ 𝒫 𝒫 dom 𝐹 )