Step |
Hyp |
Ref |
Expression |
1 |
|
psubspset.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
psubspset.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
psubspset.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
psubspset.s |
⊢ 𝑆 = ( PSubSp ‘ 𝐾 ) |
5 |
1 2 3 4
|
psubspset |
⊢ ( 𝐾 ∈ 𝐷 → 𝑆 = { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑥 ∀ 𝑞 ∈ 𝑥 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑥 ) ) } ) |
6 |
5
|
eleq2d |
⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑆 ↔ 𝑋 ∈ { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑥 ∀ 𝑞 ∈ 𝑥 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑥 ) ) } ) ) |
7 |
3
|
fvexi |
⊢ 𝐴 ∈ V |
8 |
7
|
ssex |
⊢ ( 𝑋 ⊆ 𝐴 → 𝑋 ∈ V ) |
9 |
8
|
adantr |
⊢ ( ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑋 ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑋 ) ) → 𝑋 ∈ V ) |
10 |
|
sseq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴 ) ) |
11 |
|
eleq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑟 ∈ 𝑥 ↔ 𝑟 ∈ 𝑋 ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑥 ) ↔ ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑋 ) ) ) |
13 |
12
|
ralbidv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑥 ) ↔ ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑋 ) ) ) |
14 |
13
|
raleqbi1dv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑞 ∈ 𝑥 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑥 ) ↔ ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑋 ) ) ) |
15 |
14
|
raleqbi1dv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑝 ∈ 𝑥 ∀ 𝑞 ∈ 𝑥 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑥 ) ↔ ∀ 𝑝 ∈ 𝑋 ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑋 ) ) ) |
16 |
10 15
|
anbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑥 ∀ 𝑞 ∈ 𝑥 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑥 ) ) ↔ ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑋 ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑋 ) ) ) ) |
17 |
9 16
|
elab3 |
⊢ ( 𝑋 ∈ { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑥 ∀ 𝑞 ∈ 𝑥 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑥 ) ) } ↔ ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑋 ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑋 ) ) ) |
18 |
6 17
|
bitrdi |
⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑆 ↔ ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝑋 ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) → 𝑟 ∈ 𝑋 ) ) ) ) |