Step |
Hyp |
Ref |
Expression |
1 |
|
gneispace.a |
⊢ 𝐴 = { 𝑓 ∣ ( 𝑓 : dom 𝑓 ⟶ ( 𝒫 ( 𝒫 dom 𝑓 ∖ { ∅ } ) ∖ { ∅ } ) ∧ ∀ 𝑝 ∈ dom 𝑓 ∀ 𝑛 ∈ ( 𝑓 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝑓 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝑓 ‘ 𝑝 ) ) ) ) } |
2 |
|
elex |
⊢ ( 𝐹 ∈ 𝐴 → 𝐹 ∈ V ) |
3 |
1
|
gneispace |
⊢ ( 𝐹 ∈ V → ( 𝐹 ∈ 𝐴 ↔ ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹 ‘ 𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) ) ) ) |
4 |
2 3
|
syl |
⊢ ( 𝐹 ∈ 𝐴 → ( 𝐹 ∈ 𝐴 ↔ ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹 ‘ 𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) ) ) ) |
5 |
4
|
ibi |
⊢ ( 𝐹 ∈ 𝐴 → ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹 ‘ 𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) ) ) |
6 |
|
simp1 |
⊢ ( ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹 ‘ 𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) ) → Fun 𝐹 ) |
7 |
6
|
funfnd |
⊢ ( ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹 ‘ 𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) ) → 𝐹 Fn dom 𝐹 ) |
8 |
|
simp2 |
⊢ ( ( Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀ 𝑝 ∈ dom 𝐹 ( ( 𝐹 ‘ 𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) ) → 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 ( ( 𝐹 ‘ 𝑝 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐹 ‘ 𝑝 ) ( 𝑝 ∈ 𝑛 ∧ ∀ 𝑠 ∈ 𝒫 dom 𝐹 ( 𝑛 ⊆ 𝑠 → 𝑠 ∈ ( 𝐹 ‘ 𝑝 ) ) ) ) ) → 𝐹 : dom 𝐹 ⟶ 𝒫 𝒫 dom 𝐹 ) |
11 |
5 10
|
syl |
⊢ ( 𝐹 ∈ 𝐴 → 𝐹 : dom 𝐹 ⟶ 𝒫 𝒫 dom 𝐹 ) |