Step |
Hyp |
Ref |
Expression |
1 |
|
ptbas.1 |
⊢ 𝐵 = { 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 ) ) } |
2 |
|
ptbasfi.2 |
⊢ 𝑋 = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) |
3 |
1
|
elpt |
⊢ ( 𝑠 ∈ 𝐵 ↔ ∃ ℎ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ) ) |
4 |
|
df-3an |
⊢ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
5 |
|
simprr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) |
6 |
|
disjdif2 |
⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → ( 𝐴 ∖ 𝑚 ) = 𝐴 ) |
7 |
6
|
raleqdv |
⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
8 |
7
|
biimpac |
⊢ ( ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) |
9 |
|
ixpeq2 |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) ) |
10 |
8 9
|
syl |
⊢ ( ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) ) |
11 |
|
fveq2 |
⊢ ( 𝑛 = 𝑦 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑦 ) ) |
12 |
11
|
unieqd |
⊢ ( 𝑛 = 𝑦 → ∪ ( 𝐹 ‘ 𝑛 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) |
13 |
12
|
cbvixpv |
⊢ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) |
14 |
2 13
|
eqtri |
⊢ 𝑋 = X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) |
15 |
10 14
|
eqtr4di |
⊢ ( ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = 𝑋 ) |
16 |
5 15
|
sylan |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = 𝑋 ) |
17 |
|
ssv |
⊢ 𝑋 ⊆ V |
18 |
|
iineq1 |
⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = ∩ 𝑛 ∈ ∅ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
19 |
|
0iin |
⊢ ∩ 𝑛 ∈ ∅ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = V |
20 |
18 19
|
eqtrdi |
⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = V ) |
21 |
17 20
|
sseqtrrid |
⊢ ( ( 𝐴 ∩ 𝑚 ) = ∅ → 𝑋 ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
22 |
21
|
adantl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → 𝑋 ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
23 |
|
df-ss |
⊢ ( 𝑋 ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ↔ ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = 𝑋 ) |
24 |
22 23
|
sylib |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = 𝑋 ) |
25 |
16 24
|
eqtr4d |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) = ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ) |
26 |
|
simplll |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ) |
27 |
|
inss1 |
⊢ ( 𝐴 ∩ 𝑚 ) ⊆ 𝐴 |
28 |
|
simpr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) |
29 |
27 28
|
sselid |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 𝑛 ∈ 𝐴 ) |
30 |
|
fveq2 |
⊢ ( 𝑦 = 𝑛 → ( ℎ ‘ 𝑦 ) = ( ℎ ‘ 𝑛 ) ) |
31 |
|
fveq2 |
⊢ ( 𝑦 = 𝑛 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑛 ) ) |
32 |
30 31
|
eleq12d |
⊢ ( 𝑦 = 𝑛 → ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ↔ ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ) ) |
33 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) |
34 |
33
|
ad2antrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) |
35 |
32 34 29
|
rspcdva |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ) |
36 |
14
|
ptpjpre1 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝑛 ∈ 𝐴 ∧ ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
37 |
26 29 35 36
|
syl12anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
38 |
37
|
adantlr |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
39 |
38
|
iineq2dv |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
40 |
|
simpr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) |
41 |
|
cnvimass |
⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ dom ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) |
42 |
|
eqid |
⊢ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) = ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) |
43 |
42
|
dmmptss |
⊢ dom ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) ⊆ 𝑋 |
44 |
41 43
|
sstri |
⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ 𝑋 |
45 |
44 14
|
sseqtri |
⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) |
46 |
45
|
rgenw |
⊢ ∀ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) |
47 |
|
r19.2z |
⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ ∀ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) ) → ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) ) |
48 |
40 46 47
|
sylancl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) ) |
49 |
|
iinss |
⊢ ( ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) ) |
50 |
48 49
|
syl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ X 𝑦 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑦 ) ) |
51 |
50 14
|
sseqtrrdi |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ 𝑋 ) |
52 |
|
sseqin2 |
⊢ ( ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ 𝑋 ↔ ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
53 |
51 52
|
sylib |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
54 |
33
|
ad2antrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) |
55 |
|
ssralv |
⊢ ( ( 𝐴 ∩ 𝑚 ) ⊆ 𝐴 → ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) |
56 |
27 55
|
ax-mp |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) |
57 |
|
elssuni |
⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ( ℎ ‘ 𝑦 ) ⊆ ∪ ( 𝐹 ‘ 𝑦 ) ) |
58 |
|
iffalse |
⊢ ( ¬ 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∪ ( 𝐹 ‘ 𝑦 ) ) |
59 |
58
|
sseq2d |
⊢ ( ¬ 𝑦 = 𝑛 → ( ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ( ℎ ‘ 𝑦 ) ⊆ ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
60 |
57 59
|
syl5ibrcom |
⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ( ¬ 𝑦 = 𝑛 → ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) ) |
61 |
|
ssid |
⊢ ( ℎ ‘ 𝑦 ) ⊆ ( ℎ ‘ 𝑦 ) |
62 |
|
iftrue |
⊢ ( 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ( ℎ ‘ 𝑛 ) ) |
63 |
62 30
|
eqtr4d |
⊢ ( 𝑦 = 𝑛 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ( ℎ ‘ 𝑦 ) ) |
64 |
61 63
|
sseqtrrid |
⊢ ( 𝑦 = 𝑛 → ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
65 |
60 64
|
pm2.61d2 |
⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
66 |
65
|
ralrimivw |
⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
67 |
|
ssiin |
⊢ ( ( ℎ ‘ 𝑦 ) ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ⊆ if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
68 |
66 67
|
sylibr |
⊢ ( ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ( ℎ ‘ 𝑦 ) ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
69 |
68
|
adantl |
⊢ ( ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ∧ ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) → ( ℎ ‘ 𝑦 ) ⊆ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
70 |
62
|
equcoms |
⊢ ( 𝑛 = 𝑦 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ( ℎ ‘ 𝑛 ) ) |
71 |
|
fveq2 |
⊢ ( 𝑛 = 𝑦 → ( ℎ ‘ 𝑛 ) = ( ℎ ‘ 𝑦 ) ) |
72 |
70 71
|
eqtrd |
⊢ ( 𝑛 = 𝑦 → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ( ℎ ‘ 𝑦 ) ) |
73 |
72
|
sseq1d |
⊢ ( 𝑛 = 𝑦 → ( if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) ↔ ( ℎ ‘ 𝑦 ) ⊆ ( ℎ ‘ 𝑦 ) ) ) |
74 |
73
|
rspcev |
⊢ ( ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ∧ ( ℎ ‘ 𝑦 ) ⊆ ( ℎ ‘ 𝑦 ) ) → ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) ) |
75 |
61 74
|
mpan2 |
⊢ ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) → ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) ) |
76 |
|
iinss |
⊢ ( ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) ) |
77 |
75 76
|
syl |
⊢ ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) ) |
78 |
77
|
adantr |
⊢ ( ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ∧ ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ⊆ ( ℎ ‘ 𝑦 ) ) |
79 |
69 78
|
eqssd |
⊢ ( ( 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ∧ ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) → ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
80 |
79
|
ralimiaa |
⊢ ( ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
81 |
54 56 80
|
3syl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
82 |
|
eldifn |
⊢ ( 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) → ¬ 𝑦 ∈ 𝑚 ) |
83 |
82
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ¬ 𝑦 ∈ 𝑚 ) |
84 |
|
inss2 |
⊢ ( 𝐴 ∩ 𝑚 ) ⊆ 𝑚 |
85 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) |
86 |
84 85
|
sselid |
⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 𝑛 ∈ 𝑚 ) |
87 |
|
eleq1 |
⊢ ( 𝑦 = 𝑛 → ( 𝑦 ∈ 𝑚 ↔ 𝑛 ∈ 𝑚 ) ) |
88 |
86 87
|
syl5ibrcom |
⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑦 = 𝑛 → 𝑦 ∈ 𝑚 ) ) |
89 |
83 88
|
mtod |
⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ¬ 𝑦 = 𝑛 ) |
90 |
89 58
|
syl |
⊢ ( ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∪ ( 𝐹 ‘ 𝑦 ) ) |
91 |
90
|
iineq2dv |
⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐹 ‘ 𝑦 ) ) |
92 |
|
iinconst |
⊢ ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐹 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) |
93 |
92
|
adantr |
⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐹 ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) |
94 |
91 93
|
eqtr2d |
⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) → ∪ ( 𝐹 ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
95 |
|
eqeq1 |
⊢ ( ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → ( ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ∪ ( 𝐹 ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) ) |
96 |
94 95
|
syl5ibrcom |
⊢ ( ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ ∧ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ) → ( ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) ) |
97 |
96
|
ralimdva |
⊢ ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ → ( ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) ) |
98 |
5 97
|
mpan9 |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
99 |
|
inundif |
⊢ ( ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐴 ∖ 𝑚 ) ) = 𝐴 |
100 |
99
|
raleqi |
⊢ ( ∀ 𝑦 ∈ ( ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐴 ∖ 𝑚 ) ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
101 |
|
ralunb |
⊢ ( ∀ 𝑦 ∈ ( ( 𝐴 ∩ 𝑚 ) ∪ ( 𝐴 ∖ 𝑚 ) ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) ) |
102 |
100 101
|
bitr3i |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ ( 𝐴 ∩ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) ) |
103 |
81 98 102
|
sylanbrc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
104 |
|
ixpeq2 |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = X 𝑦 ∈ 𝐴 ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
105 |
103 104
|
syl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = X 𝑦 ∈ 𝐴 ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
106 |
|
ixpiin |
⊢ ( ( 𝐴 ∩ 𝑚 ) ≠ ∅ → X 𝑦 ∈ 𝐴 ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
107 |
106
|
adantl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → X 𝑦 ∈ 𝐴 ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
108 |
105 107
|
eqtrd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) X 𝑦 ∈ 𝐴 if ( 𝑦 = 𝑛 , ( ℎ ‘ 𝑛 ) , ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
109 |
39 53 108
|
3eqtr4rd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝐴 ∩ 𝑚 ) ≠ ∅ ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ) |
110 |
25 109
|
pm2.61dane |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) = ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ) |
111 |
|
ixpexg |
⊢ ( ∀ 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∈ V → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∈ V ) |
112 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑛 ) ∈ V |
113 |
112
|
uniex |
⊢ ∪ ( 𝐹 ‘ 𝑛 ) ∈ V |
114 |
113
|
a1i |
⊢ ( 𝑛 ∈ 𝐴 → ∪ ( 𝐹 ‘ 𝑛 ) ∈ V ) |
115 |
111 114
|
mprg |
⊢ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∈ V |
116 |
2 115
|
eqeltri |
⊢ 𝑋 ∈ V |
117 |
116
|
mptex |
⊢ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) ∈ V |
118 |
117
|
cnvex |
⊢ ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) ∈ V |
119 |
118
|
imaex |
⊢ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ V |
120 |
119
|
dfiin2 |
⊢ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = ∩ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } |
121 |
|
inteq |
⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ∩ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∩ ∅ ) |
122 |
120 121
|
eqtrid |
⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = ∩ ∅ ) |
123 |
|
int0 |
⊢ ∩ ∅ = V |
124 |
122 123
|
eqtrdi |
⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) = V ) |
125 |
124
|
ineq2d |
⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = ( 𝑋 ∩ V ) ) |
126 |
|
inv1 |
⊢ ( 𝑋 ∩ V ) = 𝑋 |
127 |
125 126
|
eqtrdi |
⊢ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = 𝑋 ) |
128 |
127
|
adantl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = 𝑋 ) |
129 |
|
snex |
⊢ { 𝑋 } ∈ V |
130 |
1
|
ptbas |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 ∈ TopBases ) |
131 |
1 2
|
ptpjpre2 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝐵 ) |
132 |
131
|
ralrimivva |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∀ 𝑘 ∈ 𝐴 ∀ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝐵 ) |
133 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) = ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) |
134 |
133
|
fmpox |
⊢ ( ∀ 𝑘 ∈ 𝐴 ∀ 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ∈ 𝐵 ↔ ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) : ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ⟶ 𝐵 ) |
135 |
132 134
|
sylib |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) : ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ⟶ 𝐵 ) |
136 |
135
|
frnd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ⊆ 𝐵 ) |
137 |
130 136
|
ssexd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ∈ V ) |
138 |
|
unexg |
⊢ ( ( { 𝑋 } ∈ V ∧ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ∈ V ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V ) |
139 |
129 137 138
|
sylancr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V ) |
140 |
|
ssfii |
⊢ ( ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
141 |
139 140
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
142 |
141
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
143 |
|
ssun1 |
⊢ { 𝑋 } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
144 |
116
|
snss |
⊢ ( 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ↔ { 𝑋 } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
145 |
143 144
|
mpbir |
⊢ 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
146 |
145
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
147 |
142 146
|
sseldd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → 𝑋 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
148 |
147
|
adantr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ ) → 𝑋 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
149 |
128 148
|
eqeltrd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } = ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
150 |
139
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V ) |
151 |
|
nfv |
⊢ Ⅎ 𝑛 ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) |
152 |
|
nfcv |
⊢ Ⅎ 𝑛 𝐴 |
153 |
|
nfcv |
⊢ Ⅎ 𝑛 ( 𝐹 ‘ 𝑘 ) |
154 |
|
nfixp1 |
⊢ Ⅎ 𝑛 X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) |
155 |
2 154
|
nfcxfr |
⊢ Ⅎ 𝑛 𝑋 |
156 |
|
nfcv |
⊢ Ⅎ 𝑛 ( 𝑤 ‘ 𝑘 ) |
157 |
155 156
|
nfmpt |
⊢ Ⅎ 𝑛 ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) |
158 |
157
|
nfcnv |
⊢ Ⅎ 𝑛 ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) |
159 |
|
nfcv |
⊢ Ⅎ 𝑛 𝑢 |
160 |
158 159
|
nfima |
⊢ Ⅎ 𝑛 ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) |
161 |
152 153 160
|
nfmpo |
⊢ Ⅎ 𝑛 ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) |
162 |
161
|
nfrn |
⊢ Ⅎ 𝑛 ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) |
163 |
162
|
nfcri |
⊢ Ⅎ 𝑛 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) |
164 |
|
df-ov |
⊢ ( 𝑛 ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ( ℎ ‘ 𝑛 ) ) = ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ‘ 〈 𝑛 , ( ℎ ‘ 𝑛 ) 〉 ) |
165 |
119
|
a1i |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ V ) |
166 |
|
fveq2 |
⊢ ( 𝑘 = 𝑛 → ( 𝑤 ‘ 𝑘 ) = ( 𝑤 ‘ 𝑛 ) ) |
167 |
166
|
mpteq2dv |
⊢ ( 𝑘 = 𝑛 → ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) = ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) ) |
168 |
167
|
cnveqd |
⊢ ( 𝑘 = 𝑛 → ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) = ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) ) |
169 |
168
|
imaeq1d |
⊢ ( 𝑘 = 𝑛 → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ 𝑢 ) ) |
170 |
|
imaeq2 |
⊢ ( 𝑢 = ( ℎ ‘ 𝑛 ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ 𝑢 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
171 |
169 170
|
sylan9eq |
⊢ ( ( 𝑘 = 𝑛 ∧ 𝑢 = ( ℎ ‘ 𝑛 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
172 |
|
fveq2 |
⊢ ( 𝑘 = 𝑛 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑛 ) ) |
173 |
171 172 133
|
ovmpox |
⊢ ( ( 𝑛 ∈ 𝐴 ∧ ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ∧ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ V ) → ( 𝑛 ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ( ℎ ‘ 𝑛 ) ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
174 |
29 35 165 173
|
syl3anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑛 ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ( ℎ ‘ 𝑛 ) ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
175 |
164 174
|
eqtr3id |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ‘ 〈 𝑛 , ( ℎ ‘ 𝑛 ) 〉 ) = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
176 |
135
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) : ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ⟶ 𝐵 ) |
177 |
176
|
ffnd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) Fn ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ) |
178 |
|
opeliunxp |
⊢ ( 〈 𝑛 , ( ℎ ‘ 𝑛 ) 〉 ∈ ∪ 𝑛 ∈ 𝐴 ( { 𝑛 } × ( 𝐹 ‘ 𝑛 ) ) ↔ ( 𝑛 ∈ 𝐴 ∧ ( ℎ ‘ 𝑛 ) ∈ ( 𝐹 ‘ 𝑛 ) ) ) |
179 |
29 35 178
|
sylanbrc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 〈 𝑛 , ( ℎ ‘ 𝑛 ) 〉 ∈ ∪ 𝑛 ∈ 𝐴 ( { 𝑛 } × ( 𝐹 ‘ 𝑛 ) ) ) |
180 |
|
sneq |
⊢ ( 𝑛 = 𝑘 → { 𝑛 } = { 𝑘 } ) |
181 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑘 ) ) |
182 |
180 181
|
xpeq12d |
⊢ ( 𝑛 = 𝑘 → ( { 𝑛 } × ( 𝐹 ‘ 𝑛 ) ) = ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ) |
183 |
182
|
cbviunv |
⊢ ∪ 𝑛 ∈ 𝐴 ( { 𝑛 } × ( 𝐹 ‘ 𝑛 ) ) = ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) |
184 |
179 183
|
eleqtrdi |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → 〈 𝑛 , ( ℎ ‘ 𝑛 ) 〉 ∈ ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ) |
185 |
|
fnfvelrn |
⊢ ( ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) Fn ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ∧ 〈 𝑛 , ( ℎ ‘ 𝑛 ) 〉 ∈ ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ‘ 〈 𝑛 , ( ℎ ‘ 𝑛 ) 〉 ) ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
186 |
177 184 185
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ‘ 〈 𝑛 , ( ℎ ‘ 𝑛 ) 〉 ) ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
187 |
175 186
|
eqeltrrd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
188 |
|
eleq1 |
⊢ ( 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) → ( 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ↔ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
189 |
187 188
|
syl5ibrcom |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ) → ( 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) → 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
190 |
189
|
ex |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) → ( 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) → 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
191 |
151 163 190
|
rexlimd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) → 𝑧 ∈ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
192 |
191
|
abssdv |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ⊆ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
193 |
|
ssun2 |
⊢ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) |
194 |
192 193
|
sstrdi |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
195 |
194
|
adantr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
196 |
|
simpr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) |
197 |
|
simplrl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → 𝑚 ∈ Fin ) |
198 |
|
ssfi |
⊢ ( ( 𝑚 ∈ Fin ∧ ( 𝐴 ∩ 𝑚 ) ⊆ 𝑚 ) → ( 𝐴 ∩ 𝑚 ) ∈ Fin ) |
199 |
197 84 198
|
sylancl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ( 𝐴 ∩ 𝑚 ) ∈ Fin ) |
200 |
|
abrexfi |
⊢ ( ( 𝐴 ∩ 𝑚 ) ∈ Fin → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ Fin ) |
201 |
199 200
|
syl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ Fin ) |
202 |
|
elfir |
⊢ ( ( ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V ∧ ( { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ⊆ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ Fin ) ) → ∩ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
203 |
150 195 196 201 202
|
syl13anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∩ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
204 |
120 203
|
eqeltrid |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
205 |
|
elssuni |
⊢ ( ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
206 |
204 205
|
syl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
207 |
|
fiuni |
⊢ ( ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ∈ V → ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) = ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
208 |
139 207
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) = ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
209 |
116
|
pwid |
⊢ 𝑋 ∈ 𝒫 𝑋 |
210 |
209
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝑋 ∈ 𝒫 𝑋 ) |
211 |
210
|
snssd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑋 } ⊆ 𝒫 𝑋 ) |
212 |
1
|
ptuni2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = ∪ 𝐵 ) |
213 |
2 212
|
eqtrid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝑋 = ∪ 𝐵 ) |
214 |
|
eqimss2 |
⊢ ( 𝑋 = ∪ 𝐵 → ∪ 𝐵 ⊆ 𝑋 ) |
215 |
213 214
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ 𝐵 ⊆ 𝑋 ) |
216 |
|
sspwuni |
⊢ ( 𝐵 ⊆ 𝒫 𝑋 ↔ ∪ 𝐵 ⊆ 𝑋 ) |
217 |
215 216
|
sylibr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 ⊆ 𝒫 𝑋 ) |
218 |
136 217
|
sstrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ⊆ 𝒫 𝑋 ) |
219 |
211 218
|
unssd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝒫 𝑋 ) |
220 |
|
sspwuni |
⊢ ( ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝒫 𝑋 ↔ ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝑋 ) |
221 |
219 220
|
sylib |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝑋 ) |
222 |
|
elssuni |
⊢ ( 𝑋 ∈ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) → 𝑋 ⊆ ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
223 |
145 222
|
mp1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝑋 ⊆ ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) |
224 |
221 223
|
eqssd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) = 𝑋 ) |
225 |
208 224
|
eqtr3d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) = 𝑋 ) |
226 |
225
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∪ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) = 𝑋 ) |
227 |
206 226
|
sseqtrd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ⊆ 𝑋 ) |
228 |
227 52
|
sylib |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) = ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) |
229 |
228 204
|
eqeltrd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) ∧ { 𝑧 ∣ ∃ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) 𝑧 = ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) } ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
230 |
149 229
|
pm2.61dane |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝑋 ∩ ∩ 𝑛 ∈ ( 𝐴 ∩ 𝑚 ) ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑛 ) ) “ ( ℎ ‘ 𝑛 ) ) ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
231 |
110 230
|
eqeltrd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) ∧ ( 𝑚 ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
232 |
231
|
rexlimdvaa |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) → ( ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) ) |
233 |
232
|
impr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
234 |
4 233
|
sylan2b |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
235 |
|
eleq1 |
⊢ ( 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) → ( 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ↔ X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) ) |
236 |
234 235
|
syl5ibrcom |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) ∧ ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) ) |
237 |
236
|
expimpd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) ) |
238 |
237
|
exlimdv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( ∃ ℎ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑦 ) ∧ ∃ 𝑚 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑚 ) ( ℎ ‘ 𝑦 ) = ∪ ( 𝐹 ‘ 𝑦 ) ) ∧ 𝑠 = X 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ) → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) ) |
239 |
3 238
|
syl5bi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( 𝑠 ∈ 𝐵 → 𝑠 ∈ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) ) |
240 |
239
|
ssrdv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 ⊆ ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |
241 |
1
|
ptbasid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∈ 𝐵 ) |
242 |
2 241
|
eqeltrid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝑋 ∈ 𝐵 ) |
243 |
242
|
snssd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → { 𝑋 } ⊆ 𝐵 ) |
244 |
243 136
|
unssd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝐵 ) |
245 |
|
fiss |
⊢ ( ( 𝐵 ∈ TopBases ∧ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ⊆ 𝐵 ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ ( fi ‘ 𝐵 ) ) |
246 |
130 244 245
|
syl2anc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ ( fi ‘ 𝐵 ) ) |
247 |
1
|
ptbasin2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ 𝐵 ) = 𝐵 ) |
248 |
246 247
|
sseqtrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ⊆ 𝐵 ) |
249 |
240 248
|
eqssd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → 𝐵 = ( fi ‘ ( { 𝑋 } ∪ ran ( 𝑘 ∈ 𝐴 , 𝑢 ∈ ( 𝐹 ‘ 𝑘 ) ↦ ( ◡ ( 𝑤 ∈ 𝑋 ↦ ( 𝑤 ‘ 𝑘 ) ) “ 𝑢 ) ) ) ) ) |