Step | Hyp | Ref
| Expression |
1 | | glbeldm.b |
. . . 4
⊢ 𝐵 = (Base‘𝐾) |
2 | | glbeldm.l |
. . . 4
⊢ ≤ =
(le‘𝐾) |
3 | | glbeldm.g |
. . . 4
⊢ 𝐺 = (glb‘𝐾) |
4 | | biid 250 |
. . . 4
⊢
((∀𝑦 ∈
𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) |
5 | | glbeldm.k |
. . . 4
⊢ (𝜑 → 𝐾 ∈ 𝑉) |
6 | 1, 2, 3, 4, 5 | glbdm 16815 |
. . 3
⊢ (𝜑 → dom 𝐺 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))}) |
7 | 6 | eleq2d 2673 |
. 2
⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})) |
8 | | raleq 3115 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦)) |
9 | | raleq 3115 |
. . . . . . . . 9
⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦)) |
10 | 9 | imbi1d 330 |
. . . . . . . 8
⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) |
11 | 10 | ralbidv 2969 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) |
12 | 8, 11 | anbi12d 743 |
. . . . . 6
⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) |
13 | 12 | reubidv 3103 |
. . . . 5
⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) |
14 | | glbeldm.p |
. . . . . 6
⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) |
15 | 14 | reubii 3105 |
. . . . 5
⊢
(∃!𝑥 ∈
𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) |
16 | 13, 15 | syl6bbr 277 |
. . . 4
⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
17 | 16 | elrab 3331 |
. . 3
⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) |
18 | | fvex 6113 |
. . . . . 6
⊢
(Base‘𝐾)
∈ V |
19 | 1, 18 | eqeltri 2684 |
. . . . 5
⊢ 𝐵 ∈ V |
20 | 19 | elpw2 4755 |
. . . 4
⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
21 | 20 | anbi1i 727 |
. . 3
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) |
22 | 17, 21 | bitri 263 |
. 2
⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))} ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) |
23 | 7, 22 | syl6bb 275 |
1
⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))) |