Step | Hyp | Ref
| Expression |
1 | | rabexg 4739 |
. . . . . . 7
⊢ (𝐽 ∈ Top → {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ V) |
2 | 1 | ad2antrr 758 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ V) |
3 | | ssrab2 3650 |
. . . . . . 7
⊢ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ⊆ 𝐽 |
4 | | elpwg 4116 |
. . . . . . 7
⊢ ({𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ V → ({𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ 𝒫 𝐽 ↔ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ⊆ 𝐽)) |
5 | 3, 4 | mpbiri 247 |
. . . . . 6
⊢ ({𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ V → {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ 𝒫 𝐽) |
6 | 2, 5 | syl 17 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ 𝒫 𝐽) |
7 | | unieq 4380 |
. . . . . . . 8
⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∪ 𝑐 = ∪
{𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) |
8 | 7 | sseq2d 3596 |
. . . . . . 7
⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑆 ⊆ ∪ 𝑐 ↔ 𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})) |
9 | | pweq 4111 |
. . . . . . . . 9
⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → 𝒫 𝑐 = 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) |
10 | 9 | ineq1d 3775 |
. . . . . . . 8
⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝒫 𝑐 ∩ Fin) = (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)) |
11 | 10 | rexeqdv 3122 |
. . . . . . 7
⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑 ↔ ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑)) |
12 | 8, 11 | imbi12d 333 |
. . . . . 6
⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ((𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) ↔ (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑))) |
13 | 12 | rspcva 3280 |
. . . . 5
⊢ (({𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ 𝒫 𝐽 ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)) → (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑)) |
14 | 6, 13 | sylan 487 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)) → (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑)) |
15 | 14 | ex 449 |
. . 3
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑))) |
16 | | cmpsub.1 |
. . . . . . . 8
⊢ 𝑋 = ∪
𝐽 |
17 | 16 | restuni 20776 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾t 𝑆)) |
18 | 17 | adantr 480 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → 𝑆 = ∪ (𝐽 ↾t 𝑆)) |
19 | 18 | eqeq1d 2612 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → (𝑆 = ∪ 𝑠 ↔ ∪ (𝐽
↾t 𝑆) =
∪ 𝑠)) |
20 | | selpw 4115 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆) ↔ 𝑠 ⊆ (𝐽 ↾t 𝑆)) |
21 | | eleq2 2677 |
. . . . . . . . . . . . . . 15
⊢ (𝑆 = ∪
𝑠 → (𝑡 ∈ 𝑆 ↔ 𝑡 ∈ ∪ 𝑠)) |
22 | | eluni 4375 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ∪ 𝑠
↔ ∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠)) |
23 | 21, 22 | syl6bb 275 |
. . . . . . . . . . . . . 14
⊢ (𝑆 = ∪
𝑠 → (𝑡 ∈ 𝑆 ↔ ∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠))) |
24 | 23 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑡 ∈ 𝑆 ↔ ∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠))) |
25 | | ssel 3562 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ⊆ (𝐽 ↾t 𝑆) → (𝑢 ∈ 𝑠 → 𝑢 ∈ (𝐽 ↾t 𝑆))) |
26 | 16 | sseq2i 3593 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑆 ⊆ 𝑋 ↔ 𝑆 ⊆ ∪ 𝐽) |
27 | | uniexg 6853 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐽 ∈ Top → ∪ 𝐽
∈ V) |
28 | | ssexg 4732 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑆 ⊆ ∪ 𝐽
∧ ∪ 𝐽 ∈ V) → 𝑆 ∈ V) |
29 | 28 | ancoms 468 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((∪ 𝐽
∈ V ∧ 𝑆 ⊆
∪ 𝐽) → 𝑆 ∈ V) |
30 | 27, 29 | sylan 487 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽)
→ 𝑆 ∈
V) |
31 | 26, 30 | sylan2b 491 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V) |
32 | | elrest 15911 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝑢 ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑤 ∈ 𝐽 𝑢 = (𝑤 ∩ 𝑆))) |
33 | 31, 32 | syldan 486 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑢 ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑤 ∈ 𝐽 𝑢 = (𝑤 ∩ 𝑆))) |
34 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑤 ∩ 𝑆) ⊆ 𝑤 |
35 | | sseq1 3589 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑢 = (𝑤 ∩ 𝑆) → (𝑢 ⊆ 𝑤 ↔ (𝑤 ∩ 𝑆) ⊆ 𝑤)) |
36 | 34, 35 | mpbiri 247 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑢 = (𝑤 ∩ 𝑆) → 𝑢 ⊆ 𝑤) |
37 | 36 | sselda 3568 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑢 = (𝑤 ∩ 𝑆) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ 𝑤) |
38 | 37 | 3ad2antl3 1218 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ 𝑤) |
39 | 38 | 3adant2 1073 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ 𝑤) |
40 | | simp12 1085 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → 𝑤 ∈ 𝐽) |
41 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑢 = (𝑤 ∩ 𝑆) → (𝑢 ∈ 𝑠 ↔ (𝑤 ∩ 𝑆) ∈ 𝑠)) |
42 | 41 | biimpa 500 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑢 = (𝑤 ∩ 𝑆) ∧ 𝑢 ∈ 𝑠) → (𝑤 ∩ 𝑆) ∈ 𝑠) |
43 | 42 | 3ad2antl3 1218 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠) → (𝑤 ∩ 𝑆) ∈ 𝑠) |
44 | 43 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → (𝑤 ∩ 𝑆) ∈ 𝑠) |
45 | | ineq1 3769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 = 𝑤 → (𝑦 ∩ 𝑆) = (𝑤 ∩ 𝑆)) |
46 | 45 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 = 𝑤 → ((𝑦 ∩ 𝑆) ∈ 𝑠 ↔ (𝑤 ∩ 𝑆) ∈ 𝑠)) |
47 | 46 | elrab 3331 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ (𝑤 ∈ 𝐽 ∧ (𝑤 ∩ 𝑆) ∈ 𝑠)) |
48 | 40, 44, 47 | sylanbrc 695 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → 𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) |
49 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 𝑤 ∈ V |
50 | | eleq2 2677 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑣 = 𝑤 → (𝑡 ∈ 𝑣 ↔ 𝑡 ∈ 𝑤)) |
51 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑣 = 𝑤 → (𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ 𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})) |
52 | 50, 51 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑣 = 𝑤 → ((𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) ↔ (𝑡 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))) |
53 | 49, 52 | spcev 3273 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑡 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})) |
54 | 39, 48, 53 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})) |
55 | 54 | 3exp 1256 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) → (𝑢 ∈ 𝑠 → (𝑡 ∈ 𝑢 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))) |
56 | 55 | rexlimdv3a 3015 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑤 ∈ 𝐽 𝑢 = (𝑤 ∩ 𝑆) → (𝑢 ∈ 𝑠 → (𝑡 ∈ 𝑢 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))) |
57 | 33, 56 | sylbid 229 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑢 ∈ (𝐽 ↾t 𝑆) → (𝑢 ∈ 𝑠 → (𝑡 ∈ 𝑢 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))) |
58 | 57 | com23 84 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑢 ∈ 𝑠 → (𝑢 ∈ (𝐽 ↾t 𝑆) → (𝑡 ∈ 𝑢 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))) |
59 | 58 | com4l 90 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 ∈ 𝑠 → (𝑢 ∈ (𝐽 ↾t 𝑆) → (𝑡 ∈ 𝑢 → ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))) |
60 | 25, 59 | sylcom 30 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ⊆ (𝐽 ↾t 𝑆) → (𝑢 ∈ 𝑠 → (𝑡 ∈ 𝑢 → ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))) |
61 | 60 | com24 93 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ⊆ (𝐽 ↾t 𝑆) → ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑡 ∈ 𝑢 → (𝑢 ∈ 𝑠 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))) |
62 | 61 | impcom 445 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾t 𝑆)) → (𝑡 ∈ 𝑢 → (𝑢 ∈ 𝑠 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))) |
63 | 62 | impd 446 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾t 𝑆)) → ((𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))) |
64 | 63 | exlimdv 1848 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾t 𝑆)) → (∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))) |
65 | 64 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))) |
66 | 24, 65 | sylbid 229 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑡 ∈ 𝑆 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))) |
67 | 66 | ex 449 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾t 𝑆)) → (𝑆 = ∪ 𝑠 → (𝑡 ∈ 𝑆 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))) |
68 | 20, 67 | sylan2b 491 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → (𝑆 = ∪ 𝑠 → (𝑡 ∈ 𝑆 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))) |
69 | 68 | imp 444 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑡 ∈ 𝑆 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))) |
70 | | eluni 4375 |
. . . . . . . . 9
⊢ (𝑡 ∈ ∪ {𝑦
∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})) |
71 | 69, 70 | syl6ibr 241 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑡 ∈ 𝑆 → 𝑡 ∈ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})) |
72 | 71 | ssrdv 3574 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → 𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) |
73 | | pm2.27 41 |
. . . . . . . . 9
⊢ (𝑆 ⊆ ∪ {𝑦
∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑)) |
74 | | elin 3758 |
. . . . . . . . . . 11
⊢ (𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin) ↔ (𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin)) |
75 | | vex 3176 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑡 ∈ V |
76 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑡 → (𝑥 = (𝑧 ∩ 𝑆) ↔ 𝑡 = (𝑧 ∩ 𝑆))) |
77 | 76 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑡 → (∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆) ↔ ∃𝑧 ∈ 𝑑 𝑡 = (𝑧 ∩ 𝑆))) |
78 | 75, 77 | elab 3319 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ↔ ∃𝑧 ∈ 𝑑 𝑡 = (𝑧 ∩ 𝑆)) |
79 | | selpw 4115 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ 𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) |
80 | | ssel 3562 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑧 ∈ 𝑑 → 𝑧 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})) |
81 | | ineq1 3769 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑧 → (𝑦 ∩ 𝑆) = (𝑧 ∩ 𝑆)) |
82 | 81 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = 𝑧 → ((𝑦 ∩ 𝑆) ∈ 𝑠 ↔ (𝑧 ∩ 𝑆) ∈ 𝑠)) |
83 | 82 | elrab 3331 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ (𝑧 ∈ 𝐽 ∧ (𝑧 ∩ 𝑆) ∈ 𝑠)) |
84 | | eleq1a 2683 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑧 ∩ 𝑆) ∈ 𝑠 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)) |
85 | 84 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑧 ∈ 𝐽 ∧ (𝑧 ∩ 𝑆) ∈ 𝑠) → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)) |
86 | 83, 85 | sylbi 206 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)) |
87 | 80, 86 | syl6 34 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠))) |
88 | 87 | 2a1d 26 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠))))) |
89 | 88 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠))))) |
90 | 79, 89 | sylanb 488 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠))))) |
91 | 90 | 3imp 1249 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠))) |
92 | 91 | rexlimdv 3012 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → (∃𝑧 ∈ 𝑑 𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)) |
93 | 78, 92 | syl5bi 231 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → (𝑡 ∈ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} → 𝑡 ∈ 𝑠)) |
94 | 93 | ssrdv 3574 |
. . . . . . . . . . . . . . 15
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ⊆ 𝑠) |
95 | | vex 3176 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑑 ∈ V |
96 | 95 | abrexex 7033 |
. . . . . . . . . . . . . . . 16
⊢ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ V |
97 | 96 | elpw 4114 |
. . . . . . . . . . . . . . 15
⊢ ({𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ 𝒫 𝑠 ↔ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ⊆ 𝑠) |
98 | 94, 97 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ 𝒫 𝑠) |
99 | | abrexfi 8149 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 ∈ Fin → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ Fin) |
100 | 99 | ad2antlr 759 |
. . . . . . . . . . . . . . 15
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ Fin) |
101 | 100 | 3adant3 1074 |
. . . . . . . . . . . . . 14
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ Fin) |
102 | 98, 101 | elind 3760 |
. . . . . . . . . . . . 13
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ (𝒫 𝑠 ∩ Fin)) |
103 | | dfss 3555 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑆 ⊆ ∪ 𝑑
↔ 𝑆 = (𝑆 ∩ ∪ 𝑑)) |
104 | 103 | biimpi 205 |
. . . . . . . . . . . . . . . 16
⊢ (𝑆 ⊆ ∪ 𝑑
→ 𝑆 = (𝑆 ∩ ∪ 𝑑)) |
105 | | uniiun 4509 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ 𝑑 =
∪ 𝑧 ∈ 𝑑 𝑧 |
106 | 105 | ineq2i 3773 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑆 ∩ ∪ 𝑑) =
(𝑆 ∩ ∪ 𝑧 ∈ 𝑑 𝑧) |
107 | | iunin2 4520 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ 𝑧 ∈ 𝑑 (𝑆 ∩ 𝑧) = (𝑆 ∩ ∪
𝑧 ∈ 𝑑 𝑧) |
108 | | incom 3767 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑆 ∩ 𝑧) = (𝑧 ∩ 𝑆) |
109 | 108 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ 𝑑 → (𝑆 ∩ 𝑧) = (𝑧 ∩ 𝑆)) |
110 | 109 | iuneq2i 4475 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ 𝑧 ∈ 𝑑 (𝑆 ∩ 𝑧) = ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆) |
111 | 106, 107,
110 | 3eqtr2i 2638 |
. . . . . . . . . . . . . . . 16
⊢ (𝑆 ∩ ∪ 𝑑) =
∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆) |
112 | 104, 111 | syl6eq 2660 |
. . . . . . . . . . . . . . 15
⊢ (𝑆 ⊆ ∪ 𝑑
→ 𝑆 = ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆)) |
113 | 112 | 3ad2ant2 1076 |
. . . . . . . . . . . . . 14
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → 𝑆 = ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆)) |
114 | 18 | ad2antrl 760 |
. . . . . . . . . . . . . . 15
⊢ ((𝑆 ⊆ ∪ 𝑑
∧ (((𝐽 ∈ Top ∧
𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → 𝑆 = ∪ (𝐽 ↾t 𝑆)) |
115 | 114 | 3adant1 1072 |
. . . . . . . . . . . . . 14
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → 𝑆 = ∪ (𝐽 ↾t 𝑆)) |
116 | | vex 3176 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑧 ∈ V |
117 | 116 | inex1 4727 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∩ 𝑆) ∈ V |
118 | 117 | dfiun2 4490 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆) = ∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} |
119 | 118 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆) = ∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)}) |
120 | 113, 115,
119 | 3eqtr3d 2652 |
. . . . . . . . . . . . 13
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → ∪ (𝐽
↾t 𝑆) =
∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)}) |
121 | | unieq 4380 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} → ∪ 𝑡 = ∪
{𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)}) |
122 | 121 | eqeq2d 2620 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} → (∪
(𝐽 ↾t
𝑆) = ∪ 𝑡
↔ ∪ (𝐽 ↾t 𝑆) = ∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)})) |
123 | 122 | rspcev 3282 |
. . . . . . . . . . . . 13
⊢ (({𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ (𝒫 𝑠 ∩ Fin) ∧ ∪ (𝐽
↾t 𝑆) =
∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)}) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪
(𝐽 ↾t
𝑆) = ∪ 𝑡) |
124 | 102, 120,
123 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡) |
125 | 124 | 3exp 1256 |
. . . . . . . . . . 11
⊢ ((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡))) |
126 | 74, 125 | sylbi 206 |
. . . . . . . . . 10
⊢ (𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin) → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡))) |
127 | 126 | rexlimiv 3009 |
. . . . . . . . 9
⊢
(∃𝑑 ∈
(𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡)) |
128 | 73, 127 | syl6 34 |
. . . . . . . 8
⊢ (𝑆 ⊆ ∪ {𝑦
∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡))) |
129 | 128 | com3r 85 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡))) |
130 | 72, 129 | mpd 15 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡)) |
131 | 130 | ex 449 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → (𝑆 = ∪ 𝑠 → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡))) |
132 | 19, 131 | sylbird 249 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → (∪
(𝐽 ↾t
𝑆) = ∪ 𝑠
→ ((𝑆 ⊆ ∪ {𝑦
∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽
↾t 𝑆) =
∪ 𝑡))) |
133 | 132 | com23 84 |
. . 3
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → (∪ (𝐽
↾t 𝑆) =
∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪
(𝐽 ↾t
𝑆) = ∪ 𝑡))) |
134 | 15, 133 | syld 46 |
. 2
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → (∪ (𝐽
↾t 𝑆) =
∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪
(𝐽 ↾t
𝑆) = ∪ 𝑡))) |
135 | 134 | ralrimdva 2952 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∀𝑠 ∈ 𝒫 (𝐽 ↾t 𝑆)(∪
(𝐽 ↾t
𝑆) = ∪ 𝑠
→ ∃𝑡 ∈
(𝒫 𝑠 ∩
Fin)∪ (𝐽 ↾t 𝑆) = ∪ 𝑡))) |