Step | Hyp | Ref
| Expression |
1 | | lubval.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐾) |
2 | | lubval.l |
. . . . 5
⊢ ≤ =
(le‘𝐾) |
3 | | lubval.u |
. . . . 5
⊢ 𝑈 = (lub‘𝐾) |
4 | | biid 251 |
. . . . 5
⊢
((∀𝑦 ∈
𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
5 | | lubval.k |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ 𝑉) |
6 | 5 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝐾 ∈ 𝑉) |
7 | 1, 2, 3, 4, 6 | lubfval 16978 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑈 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})) |
8 | 7 | fveq1d 6193 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (𝑈‘𝑆) = (((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})‘𝑆)) |
9 | | lubval.p |
. . . . . 6
⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
10 | | simpr 477 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑆 ∈ dom 𝑈) |
11 | 1, 2, 3, 9, 6, 10 | lubeu 16983 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → ∃!𝑥 ∈ 𝐵 𝜓) |
12 | | raleq 3138 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)) |
13 | | raleq 3138 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧)) |
14 | 13 | imbi1d 331 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
15 | 14 | ralbidv 2986 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
16 | 12, 15 | anbi12d 747 |
. . . . . . . . 9
⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) |
17 | 16, 9 | syl6bbr 278 |
. . . . . . . 8
⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ 𝜓)) |
18 | 17 | reubidv 3126 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
19 | 18 | elabg 3351 |
. . . . . 6
⊢ (𝑆 ∈ dom 𝑈 → (𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))} ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
20 | 19 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))} ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
21 | 11, 20 | mpbird 247 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))}) |
22 | | fvres 6207 |
. . . 4
⊢ (𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))} → (((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})‘𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))‘𝑆)) |
23 | 21, 22 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))})‘𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))‘𝑆)) |
24 | | lubval.s |
. . . . . 6
⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
25 | 24 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑆 ⊆ 𝐵) |
26 | | fvex 6201 |
. . . . . . 7
⊢
(Base‘𝐾)
∈ V |
27 | 1, 26 | eqeltri 2697 |
. . . . . 6
⊢ 𝐵 ∈ V |
28 | 27 | elpw2 4828 |
. . . . 5
⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
29 | 25, 28 | sylibr 224 |
. . . 4
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → 𝑆 ∈ 𝒫 𝐵) |
30 | 17 | riotabidv 6613 |
. . . . 5
⊢ (𝑠 = 𝑆 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) = (℩𝑥 ∈ 𝐵 𝜓)) |
31 | | eqid 2622 |
. . . . 5
⊢ (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) |
32 | | riotaex 6615 |
. . . . 5
⊢
(℩𝑥
∈ 𝐵 𝜓) ∈ V |
33 | 30, 31, 32 | fvmpt 6282 |
. . . 4
⊢ (𝑆 ∈ 𝒫 𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓)) |
34 | 29, 33 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓)) |
35 | 8, 23, 34 | 3eqtrd 2660 |
. 2
⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓)) |
36 | | ndmfv 6218 |
. . . 4
⊢ (¬
𝑆 ∈ dom 𝑈 → (𝑈‘𝑆) = ∅) |
37 | 36 | adantl 482 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝑈) → (𝑈‘𝑆) = ∅) |
38 | 1, 2, 3, 9, 5 | lubeldm 16981 |
. . . . . . 7
⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))) |
39 | 38 | biimprd 238 |
. . . . . 6
⊢ (𝜑 → ((𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) → 𝑆 ∈ dom 𝑈)) |
40 | 24, 39 | mpand 711 |
. . . . 5
⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 → 𝑆 ∈ dom 𝑈)) |
41 | 40 | con3dimp 457 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝑈) → ¬ ∃!𝑥 ∈ 𝐵 𝜓) |
42 | | riotaund 6647 |
. . . 4
⊢ (¬
∃!𝑥 ∈ 𝐵 𝜓 → (℩𝑥 ∈ 𝐵 𝜓) = ∅) |
43 | 41, 42 | syl 17 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝑈) → (℩𝑥 ∈ 𝐵 𝜓) = ∅) |
44 | 37, 43 | eqtr4d 2659 |
. 2
⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝑈) → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓)) |
45 | 35, 44 | pm2.61dan 832 |
1
⊢ (𝜑 → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓)) |