Step |
Hyp |
Ref |
Expression |
1 |
|
ptpjcn.1 |
⊢ 𝑌 = ∪ 𝐽 |
2 |
|
ptpjcn.2 |
⊢ 𝐽 = ( ∏t ‘ 𝐹 ) |
3 |
2
|
ptuni |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐽 ) |
4 |
3
|
3adant3 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ 𝐽 ) |
5 |
1 4
|
eqtr4id |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → 𝑌 = X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
6 |
5
|
mpteq1d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( 𝑥 ∈ 𝑌 ↦ ( 𝑥 ‘ 𝐼 ) ) = ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) ) |
7 |
|
pttop |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( ∏t ‘ 𝐹 ) ∈ Top ) |
8 |
7
|
3adant3 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( ∏t ‘ 𝐹 ) ∈ Top ) |
9 |
2 8
|
eqeltrid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → 𝐽 ∈ Top ) |
10 |
|
ffvelrn |
⊢ ( ( 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐼 ) ∈ Top ) |
11 |
10
|
3adant1 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐼 ) ∈ Top ) |
12 |
|
vex |
⊢ 𝑥 ∈ V |
13 |
12
|
elixp |
⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↔ ( 𝑥 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑥 ‘ 𝑘 ) ∈ ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
14 |
13
|
simprbi |
⊢ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) → ∀ 𝑘 ∈ 𝐴 ( 𝑥 ‘ 𝑘 ) ∈ ∪ ( 𝐹 ‘ 𝑘 ) ) |
15 |
|
fveq2 |
⊢ ( 𝑘 = 𝐼 → ( 𝑥 ‘ 𝑘 ) = ( 𝑥 ‘ 𝐼 ) ) |
16 |
|
fveq2 |
⊢ ( 𝑘 = 𝐼 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝐼 ) ) |
17 |
16
|
unieqd |
⊢ ( 𝑘 = 𝐼 → ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝐼 ) ) |
18 |
15 17
|
eleq12d |
⊢ ( 𝑘 = 𝐼 → ( ( 𝑥 ‘ 𝑘 ) ∈ ∪ ( 𝐹 ‘ 𝑘 ) ↔ ( 𝑥 ‘ 𝐼 ) ∈ ∪ ( 𝐹 ‘ 𝐼 ) ) ) |
19 |
18
|
rspcva |
⊢ ( ( 𝐼 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑥 ‘ 𝑘 ) ∈ ∪ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑥 ‘ 𝐼 ) ∈ ∪ ( 𝐹 ‘ 𝐼 ) ) |
20 |
14 19
|
sylan2 |
⊢ ( ( 𝐼 ∈ 𝐴 ∧ 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑥 ‘ 𝐼 ) ∈ ∪ ( 𝐹 ‘ 𝐼 ) ) |
21 |
20
|
3ad2antl3 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) ∧ 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑥 ‘ 𝐼 ) ∈ ∪ ( 𝐹 ‘ 𝐼 ) ) |
22 |
21
|
fmpttd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) : X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ⟶ ∪ ( 𝐹 ‘ 𝐼 ) ) |
23 |
5
|
feq2d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) : 𝑌 ⟶ ∪ ( 𝐹 ‘ 𝐼 ) ↔ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) : X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ⟶ ∪ ( 𝐹 ‘ 𝐼 ) ) ) |
24 |
22 23
|
mpbird |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) : 𝑌 ⟶ ∪ ( 𝐹 ‘ 𝐼 ) ) |
25 |
|
eqid |
⊢ { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } = { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } |
26 |
25
|
ptbas |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ∈ TopBases ) |
27 |
|
bastg |
⊢ ( { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ∈ TopBases → { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ⊆ ( topGen ‘ { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
28 |
26 27
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ⊆ ( topGen ‘ { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
29 |
|
ffn |
⊢ ( 𝐹 : 𝐴 ⟶ Top → 𝐹 Fn 𝐴 ) |
30 |
25
|
ptval |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ) → ( ∏t ‘ 𝐹 ) = ( topGen ‘ { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
31 |
2 30
|
syl5eq |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ) → 𝐽 = ( topGen ‘ { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
32 |
29 31
|
sylan2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐽 = ( topGen ‘ { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
33 |
28 32
|
sseqtrrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ⊆ 𝐽 ) |
34 |
33
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ⊆ 𝐽 ) |
35 |
|
eqid |
⊢ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) |
36 |
25 35
|
ptpjpre2 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → ( ◡ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) “ 𝑢 ) ∈ { 𝑤 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑤 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) |
37 |
34 36
|
sseldd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝐼 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝐼 ) ) ) → ( ◡ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) “ 𝑢 ) ∈ 𝐽 ) |
38 |
37
|
expr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝐼 ∈ 𝐴 ) → ( 𝑢 ∈ ( 𝐹 ‘ 𝐼 ) → ( ◡ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) “ 𝑢 ) ∈ 𝐽 ) ) |
39 |
38
|
ralrimiv |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ 𝐼 ∈ 𝐴 ) → ∀ 𝑢 ∈ ( 𝐹 ‘ 𝐼 ) ( ◡ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) “ 𝑢 ) ∈ 𝐽 ) |
40 |
39
|
3impa |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ∀ 𝑢 ∈ ( 𝐹 ‘ 𝐼 ) ( ◡ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) “ 𝑢 ) ∈ 𝐽 ) |
41 |
24 40
|
jca |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) : 𝑌 ⟶ ∪ ( 𝐹 ‘ 𝐼 ) ∧ ∀ 𝑢 ∈ ( 𝐹 ‘ 𝐼 ) ( ◡ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) “ 𝑢 ) ∈ 𝐽 ) ) |
42 |
|
eqid |
⊢ ∪ ( 𝐹 ‘ 𝐼 ) = ∪ ( 𝐹 ‘ 𝐼 ) |
43 |
1 42
|
iscn2 |
⊢ ( ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) ∈ ( 𝐽 Cn ( 𝐹 ‘ 𝐼 ) ) ↔ ( ( 𝐽 ∈ Top ∧ ( 𝐹 ‘ 𝐼 ) ∈ Top ) ∧ ( ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) : 𝑌 ⟶ ∪ ( 𝐹 ‘ 𝐼 ) ∧ ∀ 𝑢 ∈ ( 𝐹 ‘ 𝐼 ) ( ◡ ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) “ 𝑢 ) ∈ 𝐽 ) ) ) |
44 |
9 11 41 43
|
syl21anbrc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( 𝑥 ∈ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ↦ ( 𝑥 ‘ 𝐼 ) ) ∈ ( 𝐽 Cn ( 𝐹 ‘ 𝐼 ) ) ) |
45 |
6 44
|
eqeltrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ∧ 𝐼 ∈ 𝐴 ) → ( 𝑥 ∈ 𝑌 ↦ ( 𝑥 ‘ 𝐼 ) ) ∈ ( 𝐽 Cn ( 𝐹 ‘ 𝐼 ) ) ) |