Step |
Hyp |
Ref |
Expression |
1 |
|
ptcmp.1 |
⊢ 𝑆 = ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) |
2 |
|
ptcmp.2 |
⊢ 𝑋 = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) |
3 |
|
ptcmp.3 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
ptcmp.4 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ Comp ) |
5 |
|
ptcmp.5 |
⊢ ( 𝜑 → 𝑋 ∈ ( UFL ∩ dom card ) ) |
6 |
4
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
7 |
|
eqid |
⊢ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } = { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } |
8 |
7
|
ptval |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ) → ( ∏t ‘ 𝐹 ) = ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
9 |
3 6 8
|
syl2anc |
⊢ ( 𝜑 → ( ∏t ‘ 𝐹 ) = ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
10 |
|
cmptop |
⊢ ( 𝑥 ∈ Comp → 𝑥 ∈ Top ) |
11 |
10
|
ssriv |
⊢ Comp ⊆ Top |
12 |
|
fss |
⊢ ( ( 𝐹 : 𝐴 ⟶ Comp ∧ Comp ⊆ Top ) → 𝐹 : 𝐴 ⟶ Top ) |
13 |
4 11 12
|
sylancl |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ Top ) |
14 |
7 2
|
ptbasfi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
15 |
3 13 14
|
syl2anc |
⊢ ( 𝜑 → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
16 |
|
uncom |
⊢ ( ran 𝑆 ∪ { 𝑋 } ) = ( { 𝑋 } ∪ ran 𝑆 ) |
17 |
1
|
rneqi |
⊢ ran 𝑆 = ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) |
18 |
17
|
uneq2i |
⊢ ( { 𝑋 } ∪ ran 𝑆 ) = ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
19 |
16 18
|
eqtri |
⊢ ( ran 𝑆 ∪ { 𝑋 } ) = ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
20 |
19
|
fveq2i |
⊢ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
21 |
15 20
|
eqtr4di |
⊢ ( 𝜑 → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } = ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) |
22 |
21
|
fveq2d |
⊢ ( 𝜑 → ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) = ( topGen ‘ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) ) |
23 |
9 22
|
eqtrd |
⊢ ( 𝜑 → ( ∏t ‘ 𝐹 ) = ( topGen ‘ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) ) |
24 |
23
|
unieqd |
⊢ ( 𝜑 → ∪ ( ∏t ‘ 𝐹 ) = ∪ ( topGen ‘ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) ) |
25 |
|
fibas |
⊢ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ∈ TopBases |
26 |
|
unitg |
⊢ ( ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ∈ TopBases → ∪ ( topGen ‘ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) = ∪ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) |
27 |
25 26
|
ax-mp |
⊢ ∪ ( topGen ‘ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) = ∪ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) |
28 |
24 27
|
eqtrdi |
⊢ ( 𝜑 → ∪ ( ∏t ‘ 𝐹 ) = ∪ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) |
29 |
|
eqid |
⊢ ( ∏t ‘ 𝐹 ) = ( ∏t ‘ 𝐹 ) |
30 |
29
|
ptuni |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = ∪ ( ∏t ‘ 𝐹 ) ) |
31 |
3 13 30
|
syl2anc |
⊢ ( 𝜑 → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = ∪ ( ∏t ‘ 𝐹 ) ) |
32 |
2 31
|
eqtrid |
⊢ ( 𝜑 → 𝑋 = ∪ ( ∏t ‘ 𝐹 ) ) |
33 |
5
|
pwexd |
⊢ ( 𝜑 → 𝒫 𝑋 ∈ V ) |
34 |
|
eqid |
⊢ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) = ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) |
35 |
34
|
mptpreima |
⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) = { 𝑤 ∈ 𝑋 ∣ ( 𝑤 ‘ 𝑘 ) ∈ 𝑢 } |
36 |
35
|
ssrab3 |
⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ⊆ 𝑋 |
37 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) ) → 𝑋 ∈ ( UFL ∩ dom card ) ) |
38 |
|
elpw2g |
⊢ ( 𝑋 ∈ ( UFL ∩ dom card ) → ( ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝒫 𝑋 ↔ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ⊆ 𝑋 ) ) |
39 |
37 38
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝒫 𝑋 ↔ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ⊆ 𝑋 ) ) |
40 |
36 39
|
mpbiri |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝒫 𝑋 ) |
41 |
40
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 ∀ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝒫 𝑋 ) |
42 |
1
|
fmpox |
⊢ ( ∀ 𝑘 ∈ 𝐴 ∀ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝒫 𝑋 ↔ 𝑆 : ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ⟶ 𝒫 𝑋 ) |
43 |
41 42
|
sylib |
⊢ ( 𝜑 → 𝑆 : ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ⟶ 𝒫 𝑋 ) |
44 |
43
|
frnd |
⊢ ( 𝜑 → ran 𝑆 ⊆ 𝒫 𝑋 ) |
45 |
33 44
|
ssexd |
⊢ ( 𝜑 → ran 𝑆 ∈ V ) |
46 |
|
snex |
⊢ { 𝑋 } ∈ V |
47 |
|
unexg |
⊢ ( ( ran 𝑆 ∈ V ∧ { 𝑋 } ∈ V ) → ( ran 𝑆 ∪ { 𝑋 } ) ∈ V ) |
48 |
45 46 47
|
sylancl |
⊢ ( 𝜑 → ( ran 𝑆 ∪ { 𝑋 } ) ∈ V ) |
49 |
|
fiuni |
⊢ ( ( ran 𝑆 ∪ { 𝑋 } ) ∈ V → ∪ ( ran 𝑆 ∪ { 𝑋 } ) = ∪ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) |
50 |
48 49
|
syl |
⊢ ( 𝜑 → ∪ ( ran 𝑆 ∪ { 𝑋 } ) = ∪ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) |
51 |
28 32 50
|
3eqtr4d |
⊢ ( 𝜑 → 𝑋 = ∪ ( ran 𝑆 ∪ { 𝑋 } ) ) |
52 |
51 23
|
jca |
⊢ ( 𝜑 → ( 𝑋 = ∪ ( ran 𝑆 ∪ { 𝑋 } ) ∧ ( ∏t ‘ 𝐹 ) = ( topGen ‘ ( fi ‘ ( ran 𝑆 ∪ { 𝑋 } ) ) ) ) ) |