Step |
Hyp |
Ref |
Expression |
1 |
|
isbasisg |
⊢ ( 𝐵 ∈ TopBases → ( 𝐵 ∈ TopBases ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∩ 𝑦 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝑥 ∩ 𝑦 ) ) ) ) |
2 |
1
|
ibi |
⊢ ( 𝐵 ∈ TopBases → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∩ 𝑦 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝑥 ∩ 𝑦 ) ) ) |
3 |
|
ineq1 |
⊢ ( 𝑥 = 𝐶 → ( 𝑥 ∩ 𝑦 ) = ( 𝐶 ∩ 𝑦 ) ) |
4 |
3
|
pweqd |
⊢ ( 𝑥 = 𝐶 → 𝒫 ( 𝑥 ∩ 𝑦 ) = 𝒫 ( 𝐶 ∩ 𝑦 ) ) |
5 |
4
|
ineq2d |
⊢ ( 𝑥 = 𝐶 → ( 𝐵 ∩ 𝒫 ( 𝑥 ∩ 𝑦 ) ) = ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝑦 ) ) ) |
6 |
5
|
unieqd |
⊢ ( 𝑥 = 𝐶 → ∪ ( 𝐵 ∩ 𝒫 ( 𝑥 ∩ 𝑦 ) ) = ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝑦 ) ) ) |
7 |
3 6
|
sseq12d |
⊢ ( 𝑥 = 𝐶 → ( ( 𝑥 ∩ 𝑦 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝑥 ∩ 𝑦 ) ) ↔ ( 𝐶 ∩ 𝑦 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝑦 ) ) ) ) |
8 |
|
ineq2 |
⊢ ( 𝑦 = 𝐷 → ( 𝐶 ∩ 𝑦 ) = ( 𝐶 ∩ 𝐷 ) ) |
9 |
8
|
pweqd |
⊢ ( 𝑦 = 𝐷 → 𝒫 ( 𝐶 ∩ 𝑦 ) = 𝒫 ( 𝐶 ∩ 𝐷 ) ) |
10 |
9
|
ineq2d |
⊢ ( 𝑦 = 𝐷 → ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝑦 ) ) = ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝐷 ) ) ) |
11 |
10
|
unieqd |
⊢ ( 𝑦 = 𝐷 → ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝑦 ) ) = ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝐷 ) ) ) |
12 |
8 11
|
sseq12d |
⊢ ( 𝑦 = 𝐷 → ( ( 𝐶 ∩ 𝑦 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝑦 ) ) ↔ ( 𝐶 ∩ 𝐷 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝐷 ) ) ) ) |
13 |
7 12
|
rspc2v |
⊢ ( ( 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ∩ 𝑦 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝑥 ∩ 𝑦 ) ) → ( 𝐶 ∩ 𝐷 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝐷 ) ) ) ) |
14 |
2 13
|
syl5com |
⊢ ( 𝐵 ∈ TopBases → ( ( 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 ∩ 𝐷 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝐷 ) ) ) ) |
15 |
14
|
3impib |
⊢ ( ( 𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 ∩ 𝐷 ) ⊆ ∪ ( 𝐵 ∩ 𝒫 ( 𝐶 ∩ 𝐷 ) ) ) |