Step |
Hyp |
Ref |
Expression |
1 |
|
ptcldmpt.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
ptcldmpt.j |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐽 ∈ Top ) |
3 |
|
ptcldmpt.c |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ( Clsd ‘ 𝐽 ) ) |
4 |
|
nfcv |
⊢ Ⅎ 𝑙 𝐶 |
5 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑙 / 𝑘 ⦌ 𝐶 |
6 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑙 → 𝐶 = ⦋ 𝑙 / 𝑘 ⦌ 𝐶 ) |
7 |
4 5 6
|
cbvixp |
⊢ X 𝑘 ∈ 𝐴 𝐶 = X 𝑙 ∈ 𝐴 ⦋ 𝑙 / 𝑘 ⦌ 𝐶 |
8 |
2
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) : 𝐴 ⟶ Top ) |
9 |
|
nfv |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑙 ∈ 𝐴 ) |
10 |
|
nfcv |
⊢ Ⅎ 𝑘 Clsd |
11 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑙 ) |
12 |
10 11
|
nffv |
⊢ Ⅎ 𝑘 ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑙 ) ) |
13 |
5 12
|
nfel |
⊢ Ⅎ 𝑘 ⦋ 𝑙 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑙 ) ) |
14 |
9 13
|
nfim |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑙 ∈ 𝐴 ) → ⦋ 𝑙 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑙 ) ) ) |
15 |
|
eleq1w |
⊢ ( 𝑘 = 𝑙 → ( 𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴 ) ) |
16 |
15
|
anbi2d |
⊢ ( 𝑘 = 𝑙 → ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝑙 ∈ 𝐴 ) ) ) |
17 |
|
2fveq3 |
⊢ ( 𝑘 = 𝑙 → ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑘 ) ) = ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑙 ) ) ) |
18 |
6 17
|
eleq12d |
⊢ ( 𝑘 = 𝑙 → ( 𝐶 ∈ ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑘 ) ) ↔ ⦋ 𝑙 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑙 ) ) ) ) |
19 |
16 18
|
imbi12d |
⊢ ( 𝑘 = 𝑙 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑘 ) ) ) ↔ ( ( 𝜑 ∧ 𝑙 ∈ 𝐴 ) → ⦋ 𝑙 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑙 ) ) ) ) ) |
20 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ 𝐴 ) |
21 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) |
22 |
21
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ 𝐽 ∈ Top ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑘 ) = 𝐽 ) |
23 |
20 2 22
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑘 ) = 𝐽 ) |
24 |
23
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑘 ) ) = ( Clsd ‘ 𝐽 ) ) |
25 |
3 24
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑘 ) ) ) |
26 |
14 19 25
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ 𝐴 ) → ⦋ 𝑙 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ‘ 𝑙 ) ) ) |
27 |
1 8 26
|
ptcld |
⊢ ( 𝜑 → X 𝑙 ∈ 𝐴 ⦋ 𝑙 / 𝑘 ⦌ 𝐶 ∈ ( Clsd ‘ ( ∏t ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ) ) ) |
28 |
7 27
|
eqeltrid |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 𝐶 ∈ ( Clsd ‘ ( ∏t ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐽 ) ) ) ) |