Step |
Hyp |
Ref |
Expression |
1 |
|
ptcld.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
ptcld.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ Top ) |
3 |
|
ptcld.c |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
4 |
|
eqid |
⊢ ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑘 ) |
5 |
4
|
cldss |
⊢ ( 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑘 ) ) → 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
6 |
3 5
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
7 |
6
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
8 |
|
boxriin |
⊢ ( ∀ 𝑘 ∈ 𝐴 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) → X 𝑘 ∈ 𝐴 𝐶 = ( X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∩ ∩ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ) |
9 |
7 8
|
syl |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 𝐶 = ( X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∩ ∩ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ) |
10 |
|
eqid |
⊢ ( ∏t ‘ 𝐹 ) = ( ∏t ‘ 𝐹 ) |
11 |
10
|
ptuni |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( ∏t ‘ 𝐹 ) ) |
12 |
1 2 11
|
syl2anc |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( ∏t ‘ 𝐹 ) ) |
13 |
12
|
ineq1d |
⊢ ( 𝜑 → ( X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∩ ∩ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) = ( ∪ ( ∏t ‘ 𝐹 ) ∩ ∩ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ) |
14 |
|
pttop |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ Top ) → ( ∏t ‘ 𝐹 ) ∈ Top ) |
15 |
1 2 14
|
syl2anc |
⊢ ( 𝜑 → ( ∏t ‘ 𝐹 ) ∈ Top ) |
16 |
|
sseq1 |
⊢ ( 𝐶 = if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) → ( 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ↔ if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
17 |
|
sseq1 |
⊢ ( ∪ ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) → ( ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ↔ if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
18 |
|
simpl |
⊢ ( ( 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ∧ 𝑘 = 𝑥 ) → 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
19 |
|
ssidd |
⊢ ( ( 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ∧ ¬ 𝑘 = 𝑥 ) → ∪ ( 𝐹 ‘ 𝑘 ) ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
20 |
16 17 18 19
|
ifbothda |
⊢ ( 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) → if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
21 |
20
|
ralimi |
⊢ ( ∀ 𝑘 ∈ 𝐴 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) → ∀ 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
22 |
|
ss2ixp |
⊢ ( ∀ 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( 𝐹 ‘ 𝑘 ) → X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
23 |
7 21 22
|
3syl |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
25 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( ∏t ‘ 𝐹 ) ) |
26 |
24 25
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( ∏t ‘ 𝐹 ) ) |
27 |
12
|
eqcomd |
⊢ ( 𝜑 → ∪ ( ∏t ‘ 𝐹 ) = X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ) |
28 |
27
|
difeq1d |
⊢ ( 𝜑 → ( ∪ ( ∏t ‘ 𝐹 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) = ( X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∪ ( ∏t ‘ 𝐹 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) = ( X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ) |
30 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
31 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑘 ∈ 𝐴 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) |
32 |
|
boxcutc |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 𝐶 ⊆ ∪ ( 𝐹 ‘ 𝑘 ) ) → ( X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) = X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
33 |
30 31 32
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) = X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
34 |
|
ixpeq2 |
⊢ ( ∀ 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) = if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) → X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
35 |
|
fveq2 |
⊢ ( 𝑘 = 𝑥 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑥 ) ) |
36 |
35
|
unieqd |
⊢ ( 𝑘 = 𝑥 → ∪ ( 𝐹 ‘ 𝑘 ) = ∪ ( 𝐹 ‘ 𝑥 ) ) |
37 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑥 → 𝐶 = ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) |
38 |
36 37
|
difeq12d |
⊢ ( 𝑘 = 𝑥 → ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) = ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) ) |
39 |
38
|
adantl |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ 𝑘 = 𝑥 ) → ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) = ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) ) |
40 |
39
|
ifeq1da |
⊢ ( 𝑘 ∈ 𝐴 → if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) = if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
41 |
34 40
|
mprg |
⊢ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) |
42 |
41
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑘 ) ∖ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
43 |
29 33 42
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∪ ( ∏t ‘ 𝐹 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) = X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) ) |
44 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐴 ∈ 𝑉 ) |
45 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐹 : 𝐴 ⟶ Top ) |
46 |
3
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
47 |
|
nfv |
⊢ Ⅎ 𝑥 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑘 ) ) |
48 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑥 / 𝑘 ⦌ 𝐶 |
49 |
48
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑥 ) ) |
50 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑥 → ( Clsd ‘ ( 𝐹 ‘ 𝑘 ) ) = ( Clsd ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
51 |
37 50
|
eleq12d |
⊢ ( 𝑘 = 𝑥 → ( 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑥 ) ) ) ) |
52 |
47 49 51
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ 𝐴 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
53 |
46 52
|
sylib |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
54 |
53
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
55 |
|
eqid |
⊢ ∪ ( 𝐹 ‘ 𝑥 ) = ∪ ( 𝐹 ‘ 𝑥 ) |
56 |
55
|
cldopn |
⊢ ( ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( 𝐹 ‘ 𝑥 ) ) → ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) ∈ ( 𝐹 ‘ 𝑥 ) ) |
57 |
54 56
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) ∈ ( 𝐹 ‘ 𝑥 ) ) |
58 |
44 45 57
|
ptopn2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , ( ∪ ( 𝐹 ‘ 𝑥 ) ∖ ⦋ 𝑥 / 𝑘 ⦌ 𝐶 ) , ∪ ( 𝐹 ‘ 𝑘 ) ) ∈ ( ∏t ‘ 𝐹 ) ) |
59 |
43 58
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ∪ ( ∏t ‘ 𝐹 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ( ∏t ‘ 𝐹 ) ) |
60 |
|
eqid |
⊢ ∪ ( ∏t ‘ 𝐹 ) = ∪ ( ∏t ‘ 𝐹 ) |
61 |
60
|
iscld |
⊢ ( ( ∏t ‘ 𝐹 ) ∈ Top → ( X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ↔ ( X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( ∏t ‘ 𝐹 ) ∧ ( ∪ ( ∏t ‘ 𝐹 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ( ∏t ‘ 𝐹 ) ) ) ) |
62 |
15 61
|
syl |
⊢ ( 𝜑 → ( X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ↔ ( X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( ∏t ‘ 𝐹 ) ∧ ( ∪ ( ∏t ‘ 𝐹 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ( ∏t ‘ 𝐹 ) ) ) ) |
63 |
62
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ↔ ( X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ⊆ ∪ ( ∏t ‘ 𝐹 ) ∧ ( ∪ ( ∏t ‘ 𝐹 ) ∖ X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ( ∏t ‘ 𝐹 ) ) ) ) |
64 |
26 59 63
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ) |
65 |
64
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ) |
66 |
60
|
riincld |
⊢ ( ( ( ∏t ‘ 𝐹 ) ∈ Top ∧ ∀ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ) → ( ∪ ( ∏t ‘ 𝐹 ) ∩ ∩ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ) |
67 |
15 65 66
|
syl2anc |
⊢ ( 𝜑 → ( ∪ ( ∏t ‘ 𝐹 ) ∩ ∩ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ) |
68 |
13 67
|
eqeltrd |
⊢ ( 𝜑 → ( X 𝑘 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑘 ) ∩ ∩ 𝑥 ∈ 𝐴 X 𝑘 ∈ 𝐴 if ( 𝑘 = 𝑥 , 𝐶 , ∪ ( 𝐹 ‘ 𝑘 ) ) ) ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ) |
69 |
9 68
|
eqeltrd |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 𝐶 ∈ ( Clsd ‘ ( ∏t ‘ 𝐹 ) ) ) |