Step | Hyp | Ref
| Expression |
1 | | ssrab2 3650 |
. . . 4
⊢ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝐽 |
2 | | opnfbas.1 |
. . . . . 6
⊢ 𝑋 = ∪
𝐽 |
3 | 2 | eqimss2i 3623 |
. . . . 5
⊢ ∪ 𝐽
⊆ 𝑋 |
4 | | sspwuni 4547 |
. . . . 5
⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽
⊆ 𝑋) |
5 | 3, 4 | mpbir 220 |
. . . 4
⊢ 𝐽 ⊆ 𝒫 𝑋 |
6 | 1, 5 | sstri 3577 |
. . 3
⊢ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋 |
7 | 6 | a1i 11 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋) |
8 | 2 | topopn 20536 |
. . . . . . 7
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
9 | 8 | anim1i 590 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋)) |
10 | 9 | 3adant3 1074 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋)) |
11 | | sseq2 3590 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋)) |
12 | 11 | elrab 3331 |
. . . . 5
⊢ (𝑋 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ (𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋)) |
13 | 10, 12 | sylibr 223 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝑋 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) |
14 | | ne0i 3880 |
. . . 4
⊢ (𝑋 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅) |
15 | 13, 14 | syl 17 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅) |
16 | | ss0 3926 |
. . . . . . 7
⊢ (𝑆 ⊆ ∅ → 𝑆 = ∅) |
17 | 16 | necon3ai 2807 |
. . . . . 6
⊢ (𝑆 ≠ ∅ → ¬ 𝑆 ⊆
∅) |
18 | 17 | 3ad2ant3 1077 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ¬ 𝑆 ⊆
∅) |
19 | 18 | intnand 953 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ¬ (∅ ∈
𝐽 ∧ 𝑆 ⊆ ∅)) |
20 | | df-nel 2783 |
. . . . 5
⊢ (∅
∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ ¬ ∅ ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) |
21 | | sseq2 3590 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ ∅)) |
22 | 21 | elrab 3331 |
. . . . . 6
⊢ (∅
∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ (∅ ∈ 𝐽 ∧ 𝑆 ⊆ ∅)) |
23 | 22 | notbii 309 |
. . . . 5
⊢ (¬
∅ ∈ {𝑥 ∈
𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ ¬ (∅ ∈ 𝐽 ∧ 𝑆 ⊆ ∅)) |
24 | 20, 23 | bitr2i 264 |
. . . 4
⊢ (¬
(∅ ∈ 𝐽 ∧
𝑆 ⊆ ∅) ↔
∅ ∉ {𝑥 ∈
𝐽 ∣ 𝑆 ⊆ 𝑥}) |
25 | 19, 24 | sylib 207 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) |
26 | | sseq2 3590 |
. . . . . . 7
⊢ (𝑥 = 𝑟 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑟)) |
27 | 26 | elrab 3331 |
. . . . . 6
⊢ (𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ (𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟)) |
28 | | sseq2 3590 |
. . . . . . 7
⊢ (𝑥 = 𝑠 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑠)) |
29 | 28 | elrab 3331 |
. . . . . 6
⊢ (𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠)) |
30 | 27, 29 | anbi12i 729 |
. . . . 5
⊢ ((𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ 𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) ↔ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) |
31 | | simpl 472 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → 𝐽 ∈ Top) |
32 | | simprll 798 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → 𝑟 ∈ 𝐽) |
33 | | simprrl 800 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → 𝑠 ∈ 𝐽) |
34 | | inopn 20529 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → (𝑟 ∩ 𝑠) ∈ 𝐽) |
35 | 31, 32, 33, 34 | syl3anc 1318 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → (𝑟 ∩ 𝑠) ∈ 𝐽) |
36 | | ssin 3797 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ⊆ 𝑟 ∧ 𝑆 ⊆ 𝑠) ↔ 𝑆 ⊆ (𝑟 ∩ 𝑠)) |
37 | 36 | biimpi 205 |
. . . . . . . . . . . 12
⊢ ((𝑆 ⊆ 𝑟 ∧ 𝑆 ⊆ 𝑠) → 𝑆 ⊆ (𝑟 ∩ 𝑠)) |
38 | 37 | ad2ant2l 778 |
. . . . . . . . . . 11
⊢ (((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠)) → 𝑆 ⊆ (𝑟 ∩ 𝑠)) |
39 | 38 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → 𝑆 ⊆ (𝑟 ∩ 𝑠)) |
40 | 35, 39 | jca 553 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ 𝑆 ⊆ (𝑟 ∩ 𝑠))) |
41 | 40 | 3ad2antl1 1216 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ 𝑆 ⊆ (𝑟 ∩ 𝑠))) |
42 | | sseq2 3590 |
. . . . . . . . 9
⊢ (𝑥 = (𝑟 ∩ 𝑠) → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ (𝑟 ∩ 𝑠))) |
43 | 42 | elrab 3331 |
. . . . . . . 8
⊢ ((𝑟 ∩ 𝑠) ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ 𝑆 ⊆ (𝑟 ∩ 𝑠))) |
44 | 41, 43 | sylibr 223 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → (𝑟 ∩ 𝑠) ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) |
45 | | ssid 3587 |
. . . . . . 7
⊢ (𝑟 ∩ 𝑠) ⊆ (𝑟 ∩ 𝑠) |
46 | | sseq1 3589 |
. . . . . . . 8
⊢ (𝑡 = (𝑟 ∩ 𝑠) → (𝑡 ⊆ (𝑟 ∩ 𝑠) ↔ (𝑟 ∩ 𝑠) ⊆ (𝑟 ∩ 𝑠))) |
47 | 46 | rspcev 3282 |
. . . . . . 7
⊢ (((𝑟 ∩ 𝑠) ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ (𝑟 ∩ 𝑠) ⊆ (𝑟 ∩ 𝑠)) → ∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)) |
48 | 44, 45, 47 | sylancl 693 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → ∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)) |
49 | 48 | ex 449 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠)) → ∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))) |
50 | 30, 49 | syl5bi 231 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ 𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) → ∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))) |
51 | 50 | ralrimivv 2953 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)) |
52 | 15, 25, 51 | 3jca 1235 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))) |
53 | | isfbas2 21449 |
. . . 4
⊢ (𝑋 ∈ 𝐽 → ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))))) |
54 | 8, 53 | syl 17 |
. . 3
⊢ (𝐽 ∈ Top → ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))))) |
55 | 54 | 3ad2ant1 1075 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))))) |
56 | 7, 52, 55 | mpbir2and 959 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋)) |