Step | Hyp | Ref
| Expression |
1 | | dfrex2 2979 |
. . . . . . . . . . 11
⊢
(∃𝑛 ∈
(𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 ↔ ¬ ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛) |
2 | 1 | ralbii 2963 |
. . . . . . . . . 10
⊢
(∀𝑠 ∈
𝑡 ∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 ↔ ∀𝑠 ∈ 𝑡 ¬ ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛) |
3 | | ralnex 2975 |
. . . . . . . . . 10
⊢
(∀𝑠 ∈
𝑡 ¬ ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛 ↔ ¬ ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛) |
4 | 2, 3 | bitr2i 264 |
. . . . . . . . 9
⊢ (¬
∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛 ↔ ∀𝑠 ∈ 𝑡 ∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛) |
5 | | elin 3758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ↔ (𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin)) |
6 | | elpwi 4117 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) → 𝑛 ⊆ (𝑢 ∪ {𝑠})) |
7 | 6 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → 𝑛 ⊆ (𝑢 ∪ {𝑠})) |
8 | | uncom 3719 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑢 ∪ {𝑠}) = ({𝑠} ∪ 𝑢) |
9 | 7, 8 | syl6sseq 3614 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → 𝑛 ⊆ ({𝑠} ∪ 𝑢)) |
10 | | ssundif 4004 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ⊆ ({𝑠} ∪ 𝑢) ↔ (𝑛 ∖ {𝑠}) ⊆ 𝑢) |
11 | 9, 10 | sylib 207 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → (𝑛 ∖ {𝑠}) ⊆ 𝑢) |
12 | | diffi 8077 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ Fin → (𝑛 ∖ {𝑠}) ∈ Fin) |
13 | 12 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → (𝑛 ∖ {𝑠}) ∈ Fin) |
14 | 11, 13 | jca 553 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ 𝒫 (𝑢 ∪ {𝑠}) ∧ 𝑛 ∈ Fin) → ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin)) |
15 | 5, 14 | sylbi 206 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) → ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin)) |
16 | 15 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛) → ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin)) |
17 | 16 | ad2antll 761 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin)) |
18 | | elin 3758 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∖ {𝑠}) ∈ (𝒫 𝑢 ∩ Fin) ↔ ((𝑛 ∖ {𝑠}) ∈ 𝒫 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin)) |
19 | | vex 3176 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑢 ∈ V |
20 | 19 | elpw2 4755 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∖ {𝑠}) ∈ 𝒫 𝑢 ↔ (𝑛 ∖ {𝑠}) ⊆ 𝑢) |
21 | 20 | anbi1i 727 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑛 ∖ {𝑠}) ∈ 𝒫 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin) ↔ ((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin)) |
22 | 18, 21 | bitr2i 264 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 ∖ {𝑠}) ⊆ 𝑢 ∧ (𝑛 ∖ {𝑠}) ∈ Fin) ↔ (𝑛 ∖ {𝑠}) ∈ (𝒫 𝑢 ∩ Fin)) |
23 | 17, 22 | sylib 207 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → (𝑛 ∖ {𝑠}) ∈ (𝒫 𝑢 ∩ Fin)) |
24 | | simprrr 801 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑋 = ∪ 𝑛) |
25 | | eldif 3550 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (𝑛 ∖ {𝑠}) ↔ (𝑥 ∈ 𝑛 ∧ ¬ 𝑥 ∈ {𝑠})) |
26 | 25 | simplbi2 653 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ 𝑛 → (¬ 𝑥 ∈ {𝑠} → 𝑥 ∈ (𝑛 ∖ {𝑠}))) |
27 | | elun 3715 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) ↔ (𝑥 ∈ (𝑛 ∖ {𝑠}) ∨ 𝑥 ∈ {𝑠})) |
28 | | orcom 401 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ {𝑠} ∨ 𝑥 ∈ (𝑛 ∖ {𝑠})) ↔ (𝑥 ∈ (𝑛 ∖ {𝑠}) ∨ 𝑥 ∈ {𝑠})) |
29 | 27, 28 | bitr4i 266 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) ↔ (𝑥 ∈ {𝑠} ∨ 𝑥 ∈ (𝑛 ∖ {𝑠}))) |
30 | | df-or 384 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ {𝑠} ∨ 𝑥 ∈ (𝑛 ∖ {𝑠})) ↔ (¬ 𝑥 ∈ {𝑠} → 𝑥 ∈ (𝑛 ∖ {𝑠}))) |
31 | 29, 30 | bitr2i 264 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((¬
𝑥 ∈ {𝑠} → 𝑥 ∈ (𝑛 ∖ {𝑠})) ↔ 𝑥 ∈ ((𝑛 ∖ {𝑠}) ∪ {𝑠})) |
32 | 26, 31 | sylib 207 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ 𝑛 → 𝑥 ∈ ((𝑛 ∖ {𝑠}) ∪ {𝑠})) |
33 | 32 | ssriv 3572 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑛 ⊆ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) |
34 | | uniss 4394 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ⊆ ((𝑛 ∖ {𝑠}) ∪ {𝑠}) → ∪ 𝑛 ⊆ ∪ ((𝑛
∖ {𝑠}) ∪ {𝑠})) |
35 | 33, 34 | mp1i 13 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑛
⊆ ∪ ((𝑛 ∖ {𝑠}) ∪ {𝑠})) |
36 | | uniun 4392 |
. . . . . . . . . . . . . . . . . . 19
⊢ ∪ ((𝑛
∖ {𝑠}) ∪ {𝑠}) = (∪ (𝑛
∖ {𝑠}) ∪ ∪ {𝑠}) |
37 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑠 ∈ V |
38 | 37 | unisn 4387 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ {𝑠}
= 𝑠 |
39 | 38 | uneq2i 3726 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∪ (𝑛
∖ {𝑠}) ∪ ∪ {𝑠})
= (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠) |
40 | 36, 39 | eqtri 2632 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ ((𝑛
∖ {𝑠}) ∪ {𝑠}) = (∪ (𝑛
∖ {𝑠}) ∪ 𝑠) |
41 | 35, 40 | syl6sseq 3614 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑛
⊆ (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠)) |
42 | 24, 41 | eqsstrd 3602 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑋 ⊆ (∪
(𝑛 ∖ {𝑠}) ∪ 𝑠)) |
43 | | difss 3699 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∖ {𝑠}) ⊆ 𝑛 |
44 | 43 | unissi 4397 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ (𝑛
∖ {𝑠}) ⊆ ∪ 𝑛 |
45 | | sseq2 3590 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑋 = ∪
𝑛 → (∪ (𝑛
∖ {𝑠}) ⊆ 𝑋 ↔ ∪ (𝑛
∖ {𝑠}) ⊆ ∪ 𝑛)) |
46 | 44, 45 | mpbiri 247 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑋 = ∪
𝑛 → ∪ (𝑛
∖ {𝑠}) ⊆ 𝑋) |
47 | 46 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛) → ∪ (𝑛
∖ {𝑠}) ⊆ 𝑋) |
48 | 47 | ad2antll 761 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ (𝑛
∖ {𝑠}) ⊆ 𝑋) |
49 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(𝒫 𝑥 ∩
Fin) ⊆ 𝒫 𝑥 |
50 | 49 | sseli 3564 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑡 ∈ (𝒫 𝑥 ∩ Fin) → 𝑡 ∈ 𝒫 𝑥) |
51 | 50 | elpwid 4118 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 ∈ (𝒫 𝑥 ∩ Fin) → 𝑡 ⊆ 𝑥) |
52 | 51 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩
𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢) → 𝑡 ⊆ 𝑥) |
53 | 52 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑡 ⊆ 𝑥) |
54 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑠 ∈ 𝑡) |
55 | 53, 54 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑠 ∈ 𝑥) |
56 | | elssuni 4403 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ∈ 𝑥 → 𝑠 ⊆ ∪ 𝑥) |
57 | 55, 56 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑠 ⊆ ∪ 𝑥) |
58 | | fibas 20592 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(fi‘𝑥) ∈
TopBases |
59 | | unitg 20582 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((fi‘𝑥) ∈
TopBases → ∪ (topGen‘(fi‘𝑥)) = ∪ (fi‘𝑥)) |
60 | 58, 59 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ (topGen‘(fi‘𝑥)) = ∪
(fi‘𝑥)) |
61 | | unieq 4380 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐽 = (topGen‘(fi‘𝑥)) → ∪ 𝐽 =
∪ (topGen‘(fi‘𝑥))) |
62 | 61 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) → ∪ 𝐽 =
∪ (topGen‘(fi‘𝑥))) |
63 | 62 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝐽 =
∪ (topGen‘(fi‘𝑥))) |
64 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑥 ∈ V |
65 | | fiuni 8217 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ V → ∪ 𝑥 =
∪ (fi‘𝑥)) |
66 | 64, 65 | mp1i 13 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑥 =
∪ (fi‘𝑥)) |
67 | 60, 63, 66 | 3eqtr4rd 2655 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑥 =
∪ 𝐽) |
68 | | alexsubALT.1 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑋 = ∪
𝐽 |
69 | 67, 68 | syl6eqr 2662 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∪ 𝑥 =
𝑋) |
70 | 57, 69 | sseqtrd 3604 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑠 ⊆ 𝑋) |
71 | 48, 70 | unssd 3751 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → (∪ (𝑛
∖ {𝑠}) ∪ 𝑠) ⊆ 𝑋) |
72 | 42, 71 | eqssd 3585 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → 𝑋 = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠)) |
73 | | unieq 4380 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = (𝑛 ∖ {𝑠}) → ∪ 𝑚 = ∪
(𝑛 ∖ {𝑠})) |
74 | 73 | uneq1d 3728 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = (𝑛 ∖ {𝑠}) → (∪ 𝑚 ∪ 𝑠) = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠)) |
75 | 74 | eqeq2d 2620 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝑛 ∖ {𝑠}) → (𝑋 = (∪ 𝑚 ∪ 𝑠) ↔ 𝑋 = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠))) |
76 | 75 | rspcev 3282 |
. . . . . . . . . . . . . . 15
⊢ (((𝑛 ∖ {𝑠}) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = (∪ (𝑛 ∖ {𝑠}) ∪ 𝑠)) → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠)) |
77 | 23, 72, 76 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑠 ∈ 𝑡 ∧ (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛))) → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠)) |
78 | 77 | expr 641 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ 𝑠 ∈ 𝑡) → ((𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ∧ 𝑋 = ∪ 𝑛) → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠))) |
79 | 78 | expd 451 |
. . . . . . . . . . . 12
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ 𝑠 ∈ 𝑡) → (𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) → (𝑋 = ∪ 𝑛 → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠)))) |
80 | 79 | rexlimdv 3012 |
. . . . . . . . . . 11
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ 𝑠 ∈ 𝑡) → (∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 → ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠))) |
81 | 80 | ralimdva 2945 |
. . . . . . . . . 10
⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∀𝑠 ∈ 𝑡 ∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 → ∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠))) |
82 | | inss2 3796 |
. . . . . . . . . . . . . . 15
⊢
(𝒫 𝑥 ∩
Fin) ⊆ Fin |
83 | 82 | sseli 3564 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ (𝒫 𝑥 ∩ Fin) → 𝑡 ∈ Fin) |
84 | 83 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩
𝑡) → 𝑡 ∈ Fin) |
85 | | unieq 4380 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = (𝑓‘𝑠) → ∪ 𝑚 = ∪
(𝑓‘𝑠)) |
86 | 85 | uneq1d 3728 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝑓‘𝑠) → (∪ 𝑚 ∪ 𝑠) = (∪ (𝑓‘𝑠) ∪ 𝑠)) |
87 | 86 | eqeq2d 2620 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (𝑓‘𝑠) → (𝑋 = (∪ 𝑚 ∪ 𝑠) ↔ 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) |
88 | 87 | ac6sfi 8089 |
. . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ Fin ∧ ∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠)) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) |
89 | 88 | ex 449 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ Fin →
(∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)))) |
90 | 84, 89 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩
𝑡) → (∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)))) |
91 | 90 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩
𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → (∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)))) |
92 | 91 | ad2antrl 760 |
. . . . . . . . . 10
⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∀𝑠 ∈ 𝑡 ∃𝑚 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = (∪ 𝑚 ∪ 𝑠) → ∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)))) |
93 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ 𝑠 ∈ 𝑡) → (𝑓‘𝑠) ∈ (𝒫 𝑢 ∩ Fin)) |
94 | | elin 3758 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑓‘𝑠) ∈ (𝒫 𝑢 ∩ Fin) ↔ ((𝑓‘𝑠) ∈ 𝒫 𝑢 ∧ (𝑓‘𝑠) ∈ Fin)) |
95 | | elpwi 4117 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓‘𝑠) ∈ 𝒫 𝑢 → (𝑓‘𝑠) ⊆ 𝑢) |
96 | 95 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓‘𝑠) ∈ 𝒫 𝑢 ∧ (𝑓‘𝑠) ∈ Fin) → (𝑓‘𝑠) ⊆ 𝑢) |
97 | 94, 96 | sylbi 206 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓‘𝑠) ∈ (𝒫 𝑢 ∩ Fin) → (𝑓‘𝑠) ⊆ 𝑢) |
98 | 93, 97 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ 𝑠 ∈ 𝑡) → (𝑓‘𝑠) ⊆ 𝑢) |
99 | 98 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) → ∀𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢) |
100 | | iunss 4497 |
. . . . . . . . . . . . . . . . . 18
⊢ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢 ↔ ∀𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢) |
101 | 99, 100 | sylibr 223 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢) |
102 | 101 | ad2antrl 760 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ⊆ 𝑢) |
103 | | simplrr 797 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑤 ∈ 𝑢) |
104 | 103 | snssd 4281 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → {𝑤} ⊆ 𝑢) |
105 | 102, 104 | unssd 3751 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢) |
106 | | inss2 3796 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(𝒫 𝑢 ∩
Fin) ⊆ Fin |
107 | 106, 93 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ 𝑠 ∈ 𝑡) → (𝑓‘𝑠) ∈ Fin) |
108 | 107 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) → ∀𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin) |
109 | | iunfi 8137 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑡 ∈ Fin ∧ ∀𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin) |
110 | 84, 108, 109 | syl2an 493 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩
𝑡) ∧ 𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin)) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin) |
111 | 110 | adantlr 747 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩
𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin)) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin) |
112 | 111 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑡 ∈
(𝒫 𝑥 ∩ Fin)
∧ 𝑤 = ∩ 𝑡)
∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢) ∧ 𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin)) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin) |
113 | 112 | ad2ant2lr 780 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin) |
114 | | snfi 7923 |
. . . . . . . . . . . . . . . 16
⊢ {𝑤} ∈ Fin |
115 | | unfi 8112 |
. . . . . . . . . . . . . . . 16
⊢
((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∈ Fin ∧ {𝑤} ∈ Fin) → (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin) |
116 | 113, 114,
115 | sylancl 693 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin) |
117 | 105, 116 | jca 553 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢 ∧ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin)) |
118 | | elin 3758 |
. . . . . . . . . . . . . . 15
⊢
((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ↔ ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ 𝒫 𝑢 ∧ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin)) |
119 | 19 | elpw2 4755 |
. . . . . . . . . . . . . . . 16
⊢
((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ 𝒫 𝑢 ↔ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢) |
120 | 119 | anbi1i 727 |
. . . . . . . . . . . . . . 15
⊢
(((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ 𝒫 𝑢 ∧ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin) ↔ ((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢 ∧ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin)) |
121 | 118, 120 | bitr2i 264 |
. . . . . . . . . . . . . 14
⊢
(((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑢 ∧ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ Fin) ↔ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin)) |
122 | 117, 121 | sylib 207 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin)) |
123 | | ralnex 2975 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(∀𝑠 ∈
𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) |
124 | 123 | imbi2i 325 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) ↔ (𝑣 ∈ 𝑦 → ¬ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠))) |
125 | 124 | albii 1737 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) ↔ ∀𝑦(𝑣 ∈ 𝑦 → ¬ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠))) |
126 | | alinexa 1759 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∀𝑦(𝑣 ∈ 𝑦 → ¬ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ↔ ¬ ∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠))) |
127 | 125, 126 | bitr2i 264 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ↔ ∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠))) |
128 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑠 = 𝑧 → (𝑓‘𝑠) = (𝑓‘𝑧)) |
129 | 128 | unieqd 4382 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑠 = 𝑧 → ∪ (𝑓‘𝑠) = ∪ (𝑓‘𝑧)) |
130 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑠 = 𝑧 → 𝑠 = 𝑧) |
131 | 129, 130 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑠 = 𝑧 → (∪ (𝑓‘𝑠) ∪ 𝑠) = (∪ (𝑓‘𝑧) ∪ 𝑧)) |
132 | 131 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑠 = 𝑧 → (𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) ↔ 𝑋 = (∪ (𝑓‘𝑧) ∪ 𝑧))) |
133 | 132 | rspcv 3278 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑧 ∈ 𝑡 → (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → 𝑋 = (∪ (𝑓‘𝑧) ∪ 𝑧))) |
134 | | eleq2 2677 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑋 = (∪
(𝑓‘𝑧) ∪ 𝑧) → (𝑣 ∈ 𝑋 ↔ 𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧))) |
135 | 134 | biimpd 218 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑋 = (∪
(𝑓‘𝑧) ∪ 𝑧) → (𝑣 ∈ 𝑋 → 𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧))) |
136 | | elun 3715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧) ↔ (𝑣 ∈ ∪ (𝑓‘𝑧) ∨ 𝑣 ∈ 𝑧)) |
137 | | eluni 4375 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑣 ∈ ∪ (𝑓‘𝑧) ↔ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧))) |
138 | 137 | orbi1i 541 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑣 ∈ ∪ (𝑓‘𝑧) ∨ 𝑣 ∈ 𝑧) ↔ (∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) ∨ 𝑣 ∈ 𝑧)) |
139 | | df-or 384 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) ∨ 𝑣 ∈ 𝑧) ↔ (¬ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧)) |
140 | | alinexa 1759 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) ↔ ¬ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧))) |
141 | 140 | imbi1i 338 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧) ↔ (¬ ∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧)) |
142 | 139, 141 | bitr4i 266 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((∃𝑤(𝑣 ∈ 𝑤 ∧ 𝑤 ∈ (𝑓‘𝑧)) ∨ 𝑣 ∈ 𝑧) ↔ (∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧)) |
143 | 136, 138,
142 | 3bitri 285 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧) ↔ (∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧)) |
144 | | eleq2 2677 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑦 = 𝑤 → (𝑣 ∈ 𝑦 ↔ 𝑣 ∈ 𝑤)) |
145 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑓‘𝑠) ↔ 𝑤 ∈ (𝑓‘𝑠))) |
146 | 145 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (𝑦 = 𝑤 → (¬ 𝑦 ∈ (𝑓‘𝑠) ↔ ¬ 𝑤 ∈ (𝑓‘𝑠))) |
147 | 146 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑦 = 𝑤 → (∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠) ↔ ∀𝑠 ∈ 𝑡 ¬ 𝑤 ∈ (𝑓‘𝑠))) |
148 | 144, 147 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑦 = 𝑤 → ((𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) ↔ (𝑣 ∈ 𝑤 → ∀𝑠 ∈ 𝑡 ¬ 𝑤 ∈ (𝑓‘𝑠)))) |
149 | 148 | spv 2248 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → (𝑣 ∈ 𝑤 → ∀𝑠 ∈ 𝑡 ¬ 𝑤 ∈ (𝑓‘𝑠))) |
150 | 128 | eleq2d 2673 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑠 = 𝑧 → (𝑤 ∈ (𝑓‘𝑠) ↔ 𝑤 ∈ (𝑓‘𝑧))) |
151 | 150 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑠 = 𝑧 → (¬ 𝑤 ∈ (𝑓‘𝑠) ↔ ¬ 𝑤 ∈ (𝑓‘𝑧))) |
152 | 151 | rspcv 3278 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑧 ∈ 𝑡 → (∀𝑠 ∈ 𝑡 ¬ 𝑤 ∈ (𝑓‘𝑠) → ¬ 𝑤 ∈ (𝑓‘𝑧))) |
153 | 149, 152 | syl9r 76 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑧 ∈ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → (𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)))) |
154 | 153 | alrimdv 1844 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑧 ∈ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → ∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)))) |
155 | 154 | imim1d 80 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑧 ∈ 𝑡 → ((∀𝑤(𝑣 ∈ 𝑤 → ¬ 𝑤 ∈ (𝑓‘𝑧)) → 𝑣 ∈ 𝑧) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧))) |
156 | 143, 155 | syl5bi 231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑧 ∈ 𝑡 → (𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧))) |
157 | 156 | a1dd 48 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑧 ∈ 𝑡 → (𝑣 ∈ (∪ (𝑓‘𝑧) ∪ 𝑧) → (𝑤 = ∩ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧)))) |
158 | 135, 157 | syl9r 76 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑧 ∈ 𝑡 → (𝑋 = (∪ (𝑓‘𝑧) ∪ 𝑧) → (𝑣 ∈ 𝑋 → (𝑤 = ∩ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧))))) |
159 | 133, 158 | syld 46 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑧 ∈ 𝑡 → (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑣 ∈ 𝑋 → (𝑤 = ∩ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧))))) |
160 | 159 | com14 94 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑤 = ∩
𝑡 → (∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑣 ∈ 𝑋 → (𝑧 ∈ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧))))) |
161 | 160 | imp31 447 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (𝑧 ∈ 𝑡 → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑧))) |
162 | 161 | com23 84 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → (𝑧 ∈ 𝑡 → 𝑣 ∈ 𝑧))) |
163 | 162 | ralrimdv 2951 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → ∀𝑧 ∈ 𝑡 𝑣 ∈ 𝑧)) |
164 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 𝑣 ∈ V |
165 | 164 | elint2 4417 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑣 ∈ ∩ 𝑡
↔ ∀𝑧 ∈
𝑡 𝑣 ∈ 𝑧) |
166 | 163, 165 | syl6ibr 241 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ ∩ 𝑡)) |
167 | | eleq2 2677 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑤 = ∩
𝑡 → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ ∩ 𝑡)) |
168 | 167 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ ∩ 𝑡)) |
169 | 166, 168 | sylibrd 248 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∀𝑦(𝑣 ∈ 𝑦 → ∀𝑠 ∈ 𝑡 ¬ 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑤)) |
170 | 127, 169 | syl5bi 231 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (¬ ∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) → 𝑣 ∈ 𝑤)) |
171 | 170 | orrd 392 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) ∧ 𝑣 ∈ 𝑋) → (∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ∨ 𝑣 ∈ 𝑤)) |
172 | 171 | ex 449 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → (𝑣 ∈ 𝑋 → (∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ∨ 𝑣 ∈ 𝑤))) |
173 | | orc 399 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∃𝑠 ∈
𝑡 𝑦 ∈ (𝑓‘𝑠) → (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)) |
174 | 173 | anim2i 591 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) → (𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))) |
175 | 174 | eximi 1752 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) → ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))) |
176 | | equid 1926 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 𝑤 = 𝑤 |
177 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 𝑤 ∈ V |
178 | | equequ1 1939 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 = 𝑤 → (𝑦 = 𝑤 ↔ 𝑤 = 𝑤)) |
179 | 144, 178 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = 𝑤 → ((𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤) ↔ (𝑣 ∈ 𝑤 ∧ 𝑤 = 𝑤))) |
180 | 177, 179 | spcev 3273 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑣 ∈ 𝑤 ∧ 𝑤 = 𝑤) → ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤)) |
181 | 176, 180 | mpan2 703 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 ∈ 𝑤 → ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤)) |
182 | | olc 398 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 = 𝑤 → (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)) |
183 | 182 | anim2i 591 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤) → (𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))) |
184 | 183 | eximi 1752 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 = 𝑤) → ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))) |
185 | 181, 184 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ 𝑤 → ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))) |
186 | 175, 185 | jaoi 393 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ∨ 𝑣 ∈ 𝑤) → ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))) |
187 | | eluni 4375 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))) |
188 | | elun 3715 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ (𝑦 ∈ ∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∨ 𝑦 ∈ {𝑤})) |
189 | | eliun 4460 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ↔ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) |
190 | | velsn 4141 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑦 ∈ {𝑤} ↔ 𝑦 = 𝑤) |
191 | 189, 190 | orbi12i 542 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∨ 𝑦 ∈ {𝑤}) ↔ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)) |
192 | 188, 191 | bitri 263 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)) |
193 | 192 | anbi2i 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ∈ 𝑦 ∧ 𝑦 ∈ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) ↔ (𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))) |
194 | 193 | exbii 1764 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑦(𝑣 ∈ 𝑦 ∧ 𝑦 ∈ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) ↔ ∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤))) |
195 | 187, 194 | bitr2i 264 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∃𝑦(𝑣 ∈ 𝑦 ∧ (∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠) ∨ 𝑦 = 𝑤)) ↔ 𝑣 ∈ ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) |
196 | 186, 195 | sylib 207 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∃𝑦(𝑣 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝑡 𝑦 ∈ (𝑓‘𝑠)) ∨ 𝑣 ∈ 𝑤) → 𝑣 ∈ ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) |
197 | 172, 196 | syl6 34 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑤 = ∩
𝑡 ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → (𝑣 ∈ 𝑋 → 𝑣 ∈ ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))) |
198 | 197 | adantll 746 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩
𝑡) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → (𝑣 ∈ 𝑋 → 𝑣 ∈ ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))) |
199 | 198 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩
𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → (𝑣 ∈ 𝑋 → 𝑣 ∈ ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))) |
200 | 199 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑡 ∈
(𝒫 𝑥 ∩ Fin)
∧ 𝑤 = ∩ 𝑡)
∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → (𝑣 ∈ 𝑋 → 𝑣 ∈ ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))) |
201 | 200 | ad2ant2l 778 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (𝑣 ∈ 𝑋 → 𝑣 ∈ ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))) |
202 | 201 | ssrdv 3574 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑋 ⊆ ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) |
203 | | elun 3715 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ (𝑣 ∈ ∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∨ 𝑣 ∈ {𝑤})) |
204 | | eliun 4460 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ↔ ∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠)) |
205 | | velsn 4141 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 ∈ {𝑤} ↔ 𝑣 = 𝑤) |
206 | 204, 205 | orbi12i 542 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ∈ ∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∨ 𝑣 ∈ {𝑤}) ↔ (∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) ∨ 𝑣 = 𝑤)) |
207 | 203, 206 | bitri 263 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ↔ (∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) ∨ 𝑣 = 𝑤)) |
208 | | nfra1 2925 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) |
209 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠 𝑣 ⊆ 𝑋 |
210 | | rsp 2913 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑠 ∈
𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑠 ∈ 𝑡 → 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) |
211 | | eqimss2 3621 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑋 = (∪
(𝑓‘𝑠) ∪ 𝑠) → (∪ (𝑓‘𝑠) ∪ 𝑠) ⊆ 𝑋) |
212 | | elssuni 4403 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ ∪ (𝑓‘𝑠)) |
213 | | ssun3 3740 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 ⊆ ∪ (𝑓‘𝑠) → 𝑣 ⊆ (∪ (𝑓‘𝑠) ∪ 𝑠)) |
214 | 212, 213 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ (∪ (𝑓‘𝑠) ∪ 𝑠)) |
215 | | sstr 3576 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ⊆ (∪ (𝑓‘𝑠) ∪ 𝑠) ∧ (∪ (𝑓‘𝑠) ∪ 𝑠) ⊆ 𝑋) → 𝑣 ⊆ 𝑋) |
216 | 215 | expcom 450 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((∪ (𝑓‘𝑠) ∪ 𝑠) ⊆ 𝑋 → (𝑣 ⊆ (∪ (𝑓‘𝑠) ∪ 𝑠) → 𝑣 ⊆ 𝑋)) |
217 | 211, 214,
216 | syl2im 39 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑋 = (∪
(𝑓‘𝑠) ∪ 𝑠) → (𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ 𝑋)) |
218 | 210, 217 | syl6 34 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑠 ∈
𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (𝑠 ∈ 𝑡 → (𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ 𝑋))) |
219 | 208, 209,
218 | rexlimd 3008 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑠 ∈
𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠) → (∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ 𝑋)) |
220 | 219 | ad2antll 761 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) → 𝑣 ⊆ 𝑋)) |
221 | | elpwi 4117 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑢 ∈ 𝒫
(fi‘𝑥) → 𝑢 ⊆ (fi‘𝑥)) |
222 | 221 | ad2antrl 760 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) → 𝑢 ⊆ (fi‘𝑥)) |
223 | 222 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑢 ⊆ (fi‘𝑥)) |
224 | 223, 103 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑤 ∈ (fi‘𝑥)) |
225 | | elssuni 4403 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 ∈ (fi‘𝑥) → 𝑤 ⊆ ∪
(fi‘𝑥)) |
226 | 224, 225 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑤 ⊆ ∪
(fi‘𝑥)) |
227 | 58, 59 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∪ (topGen‘(fi‘𝑥)) = ∪
(fi‘𝑥) |
228 | 61, 227 | syl6req 2661 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐽 = (topGen‘(fi‘𝑥)) → ∪ (fi‘𝑥) = ∪ 𝐽) |
229 | 228, 68 | syl6eqr 2662 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐽 = (topGen‘(fi‘𝑥)) → ∪ (fi‘𝑥) = 𝑋) |
230 | 229 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) → ∪ (fi‘𝑥) = 𝑋) |
231 | 230 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∪
(fi‘𝑥) = 𝑋) |
232 | 226, 231 | sseqtrd 3604 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑤 ⊆ 𝑋) |
233 | | sseq1 3589 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝑤 → (𝑣 ⊆ 𝑋 ↔ 𝑤 ⊆ 𝑋)) |
234 | 232, 233 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (𝑣 = 𝑤 → 𝑣 ⊆ 𝑋)) |
235 | 220, 234 | jaod 394 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ((∃𝑠 ∈ 𝑡 𝑣 ∈ (𝑓‘𝑠) ∨ 𝑣 = 𝑤) → 𝑣 ⊆ 𝑋)) |
236 | 207, 235 | syl5bi 231 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → (𝑣 ∈ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) → 𝑣 ⊆ 𝑋)) |
237 | 236 | ralrimiv 2948 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∀𝑣 ∈ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})𝑣 ⊆ 𝑋) |
238 | | unissb 4405 |
. . . . . . . . . . . . . . 15
⊢ (∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑋 ↔ ∀𝑣 ∈ (∪
𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})𝑣 ⊆ 𝑋) |
239 | 237, 238 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ⊆ 𝑋) |
240 | 202, 239 | eqssd 3585 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → 𝑋 = ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) |
241 | | unieq 4380 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) → ∪ 𝑏 = ∪
(∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) |
242 | 241 | eqeq2d 2620 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) → (𝑋 = ∪ 𝑏 ↔ 𝑋 = ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}))) |
243 | 242 | rspcev 3282 |
. . . . . . . . . . . . 13
⊢
(((∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤}) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ∪ (∪ 𝑠 ∈ 𝑡 (𝑓‘𝑠) ∪ {𝑤})) → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏) |
244 | 122, 240,
243 | syl2anc 691 |
. . . . . . . . . . . 12
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) ∧ (𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠))) → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏) |
245 | 244 | ex 449 |
. . . . . . . . . . 11
⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → ((𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏)) |
246 | 245 | exlimdv 1848 |
. . . . . . . . . 10
⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∃𝑓(𝑓:𝑡⟶(𝒫 𝑢 ∩ Fin) ∧ ∀𝑠 ∈ 𝑡 𝑋 = (∪ (𝑓‘𝑠) ∪ 𝑠)) → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏)) |
247 | 81, 92, 246 | 3syld 58 |
. . . . . . . . 9
⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∀𝑠 ∈ 𝑡 ∃𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin)𝑋 = ∪ 𝑛 → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏)) |
248 | 4, 247 | syl5bi 231 |
. . . . . . . 8
⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (¬ ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛 → ∃𝑏 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑏)) |
249 | | dfrex2 2979 |
. . . . . . . 8
⊢
(∃𝑏 ∈
(𝒫 𝑢 ∩
Fin)𝑋 = ∪ 𝑏
↔ ¬ ∀𝑏
∈ (𝒫 𝑢 ∩
Fin) ¬ 𝑋 = ∪ 𝑏) |
250 | 248, 249 | syl6ib 240 |
. . . . . . 7
⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (¬ ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛 → ¬ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪
𝑏)) |
251 | 250 | con4d 113 |
. . . . . 6
⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) ∧ (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) ∧ 𝑤 ∈ 𝑢)) → (∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)) |
252 | 251 | exp32 629 |
. . . . 5
⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) → (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → (𝑤 ∈ 𝑢 → (∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)))) |
253 | 252 | com24 93 |
. . . 4
⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ 𝑎 ⊆ 𝑢)) → (∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑤 ∈ 𝑢 → (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)))) |
254 | 253 | exp32 629 |
. . 3
⊢ ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) → (𝑢 ∈ 𝒫 (fi‘𝑥) → (𝑎 ⊆ 𝑢 → (∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑤 ∈ 𝑢 → (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛)))))) |
255 | 254 | imp45 621 |
. 2
⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑢 ∧ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) → (𝑤 ∈ 𝑢 → (((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛))) |
256 | 255 | imp31 447 |
1
⊢
(((((𝐽 =
(topGen‘(fi‘𝑥))
∧ ∀𝑐 ∈
𝒫 𝑥(𝑋 = ∪
𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑢 ∧ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) ∧ 𝑤 ∈ 𝑢) ∧ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = ∩ 𝑡) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑦 ∈ ∪ (𝑥 ∩ 𝑢)))) → ∃𝑠 ∈ 𝑡 ∀𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = ∪ 𝑛) |