Step | Hyp | Ref
| Expression |
1 | | simpr 484 |
. . . . 5
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ⊆ 𝑌) |
2 | | sseqin2 4146 |
. . . . 5
⊢ (𝐴 ⊆ 𝑌 ↔ (𝑌 ∩ 𝐴) = 𝐴) |
3 | 1, 2 | sylib 217 |
. . . 4
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑌 ∩ 𝐴) = 𝐴) |
4 | | simpl 482 |
. . . . 5
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐿 ∈ (Fil‘𝑌)) |
5 | | id 22 |
. . . . . 6
⊢ (𝐴 ⊆ 𝑌 → 𝐴 ⊆ 𝑌) |
6 | | filtop 22914 |
. . . . . 6
⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿) |
7 | | ssexg 5242 |
. . . . . 6
⊢ ((𝐴 ⊆ 𝑌 ∧ 𝑌 ∈ 𝐿) → 𝐴 ∈ V) |
8 | 5, 6, 7 | syl2anr 596 |
. . . . 5
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ V) |
9 | 6 | adantr 480 |
. . . . 5
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝑌 ∈ 𝐿) |
10 | | elrestr 17056 |
. . . . 5
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ 𝑌 ∈ 𝐿) → (𝑌 ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
11 | 4, 8, 9, 10 | syl3anc 1369 |
. . . 4
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑌 ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
12 | 3, 11 | eqeltrrd 2840 |
. . 3
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ (𝐿 ↾t 𝐴)) |
13 | | elpwi 4539 |
. . . . 5
⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) |
14 | | vex 3426 |
. . . . . . . . . 10
⊢ 𝑢 ∈ V |
15 | 14 | inex1 5236 |
. . . . . . . . 9
⊢ (𝑢 ∩ 𝐴) ∈ V |
16 | 15 | a1i 11 |
. . . . . . . 8
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ 𝑢 ∈ 𝐿) → (𝑢 ∩ 𝐴) ∈ V) |
17 | | elrest 17055 |
. . . . . . . . . 10
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V) → (𝑦 ∈ (𝐿 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐿 𝑦 = (𝑢 ∩ 𝐴))) |
18 | 8, 17 | syldan 590 |
. . . . . . . . 9
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑦 ∈ (𝐿 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐿 𝑦 = (𝑢 ∩ 𝐴))) |
19 | 18 | adantr 480 |
. . . . . . . 8
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) → (𝑦 ∈ (𝐿 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐿 𝑦 = (𝑢 ∩ 𝐴))) |
20 | | simpr 484 |
. . . . . . . . 9
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ 𝑦 = (𝑢 ∩ 𝐴)) → 𝑦 = (𝑢 ∩ 𝐴)) |
21 | 20 | sseq1d 3948 |
. . . . . . . 8
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ 𝑦 = (𝑢 ∩ 𝐴)) → (𝑦 ⊆ 𝑥 ↔ (𝑢 ∩ 𝐴) ⊆ 𝑥)) |
22 | 16, 19, 21 | rexxfr2d 5329 |
. . . . . . 7
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) → (∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 ↔ ∃𝑢 ∈ 𝐿 (𝑢 ∩ 𝐴) ⊆ 𝑥)) |
23 | | indir 4206 |
. . . . . . . . . 10
⊢ ((𝑢 ∪ 𝑥) ∩ 𝐴) = ((𝑢 ∩ 𝐴) ∪ (𝑥 ∩ 𝐴)) |
24 | | simplr 765 |
. . . . . . . . . . . . 13
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝑥 ⊆ 𝐴) |
25 | | df-ss 3900 |
. . . . . . . . . . . . 13
⊢ (𝑥 ⊆ 𝐴 ↔ (𝑥 ∩ 𝐴) = 𝑥) |
26 | 24, 25 | sylib 217 |
. . . . . . . . . . . 12
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → (𝑥 ∩ 𝐴) = 𝑥) |
27 | 26 | uneq2d 4093 |
. . . . . . . . . . 11
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → ((𝑢 ∩ 𝐴) ∪ (𝑥 ∩ 𝐴)) = ((𝑢 ∩ 𝐴) ∪ 𝑥)) |
28 | | simprr 769 |
. . . . . . . . . . . 12
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → (𝑢 ∩ 𝐴) ⊆ 𝑥) |
29 | | ssequn1 4110 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∩ 𝐴) ⊆ 𝑥 ↔ ((𝑢 ∩ 𝐴) ∪ 𝑥) = 𝑥) |
30 | 28, 29 | sylib 217 |
. . . . . . . . . . 11
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → ((𝑢 ∩ 𝐴) ∪ 𝑥) = 𝑥) |
31 | 27, 30 | eqtrd 2778 |
. . . . . . . . . 10
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → ((𝑢 ∩ 𝐴) ∪ (𝑥 ∩ 𝐴)) = 𝑥) |
32 | 23, 31 | eqtrid 2790 |
. . . . . . . . 9
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → ((𝑢 ∪ 𝑥) ∩ 𝐴) = 𝑥) |
33 | | simplll 771 |
. . . . . . . . . 10
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝐿 ∈ (Fil‘𝑌)) |
34 | | simpllr 772 |
. . . . . . . . . . 11
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝐴 ⊆ 𝑌) |
35 | 33, 34, 8 | syl2anc 583 |
. . . . . . . . . 10
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝐴 ∈ V) |
36 | | simprl 767 |
. . . . . . . . . . 11
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝑢 ∈ 𝐿) |
37 | | filelss 22911 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝑢 ∈ 𝐿) → 𝑢 ⊆ 𝑌) |
38 | 33, 36, 37 | syl2anc 583 |
. . . . . . . . . . . 12
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝑢 ⊆ 𝑌) |
39 | 24, 34 | sstrd 3927 |
. . . . . . . . . . . 12
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝑥 ⊆ 𝑌) |
40 | 38, 39 | unssd 4116 |
. . . . . . . . . . 11
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → (𝑢 ∪ 𝑥) ⊆ 𝑌) |
41 | | ssun1 4102 |
. . . . . . . . . . . 12
⊢ 𝑢 ⊆ (𝑢 ∪ 𝑥) |
42 | 41 | a1i 11 |
. . . . . . . . . . 11
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝑢 ⊆ (𝑢 ∪ 𝑥)) |
43 | | filss 22912 |
. . . . . . . . . . 11
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∪ 𝑥) ⊆ 𝑌 ∧ 𝑢 ⊆ (𝑢 ∪ 𝑥))) → (𝑢 ∪ 𝑥) ∈ 𝐿) |
44 | 33, 36, 40, 42, 43 | syl13anc 1370 |
. . . . . . . . . 10
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → (𝑢 ∪ 𝑥) ∈ 𝐿) |
45 | | elrestr 17056 |
. . . . . . . . . 10
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ (𝑢 ∪ 𝑥) ∈ 𝐿) → ((𝑢 ∪ 𝑥) ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
46 | 33, 35, 44, 45 | syl3anc 1369 |
. . . . . . . . 9
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → ((𝑢 ∪ 𝑥) ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
47 | 32, 46 | eqeltrrd 2840 |
. . . . . . . 8
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) ∧ (𝑢 ∈ 𝐿 ∧ (𝑢 ∩ 𝐴) ⊆ 𝑥)) → 𝑥 ∈ (𝐿 ↾t 𝐴)) |
48 | 47 | rexlimdvaa 3213 |
. . . . . . 7
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) → (∃𝑢 ∈ 𝐿 (𝑢 ∩ 𝐴) ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴))) |
49 | 22, 48 | sylbid 239 |
. . . . . 6
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 ⊆ 𝐴) → (∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴))) |
50 | 49 | ex 412 |
. . . . 5
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑥 ⊆ 𝐴 → (∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)))) |
51 | 13, 50 | syl5 34 |
. . . 4
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑥 ∈ 𝒫 𝐴 → (∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)))) |
52 | 51 | ralrimiv 3106 |
. . 3
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴))) |
53 | | simpll 763 |
. . . . . 6
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ (𝑧 ∈ 𝐿 ∧ 𝑢 ∈ 𝐿)) → 𝐿 ∈ (Fil‘𝑌)) |
54 | 8 | adantr 480 |
. . . . . 6
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ (𝑧 ∈ 𝐿 ∧ 𝑢 ∈ 𝐿)) → 𝐴 ∈ V) |
55 | | filin 22913 |
. . . . . . . 8
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝑧 ∈ 𝐿 ∧ 𝑢 ∈ 𝐿) → (𝑧 ∩ 𝑢) ∈ 𝐿) |
56 | 55 | 3expb 1118 |
. . . . . . 7
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑧 ∈ 𝐿 ∧ 𝑢 ∈ 𝐿)) → (𝑧 ∩ 𝑢) ∈ 𝐿) |
57 | 56 | adantlr 711 |
. . . . . 6
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ (𝑧 ∈ 𝐿 ∧ 𝑢 ∈ 𝐿)) → (𝑧 ∩ 𝑢) ∈ 𝐿) |
58 | | elrestr 17056 |
. . . . . 6
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ (𝑧 ∩ 𝑢) ∈ 𝐿) → ((𝑧 ∩ 𝑢) ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
59 | 53, 54, 57, 58 | syl3anc 1369 |
. . . . 5
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ (𝑧 ∈ 𝐿 ∧ 𝑢 ∈ 𝐿)) → ((𝑧 ∩ 𝑢) ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
60 | 59 | ralrimivva 3114 |
. . . 4
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ∀𝑧 ∈ 𝐿 ∀𝑢 ∈ 𝐿 ((𝑧 ∩ 𝑢) ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
61 | | vex 3426 |
. . . . . . 7
⊢ 𝑧 ∈ V |
62 | 61 | inex1 5236 |
. . . . . 6
⊢ (𝑧 ∩ 𝐴) ∈ V |
63 | 62 | a1i 11 |
. . . . 5
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑧 ∈ 𝐿) → (𝑧 ∩ 𝐴) ∈ V) |
64 | | elrest 17055 |
. . . . . 6
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐿 ↾t 𝐴) ↔ ∃𝑧 ∈ 𝐿 𝑥 = (𝑧 ∩ 𝐴))) |
65 | 8, 64 | syldan 590 |
. . . . 5
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑥 ∈ (𝐿 ↾t 𝐴) ↔ ∃𝑧 ∈ 𝐿 𝑥 = (𝑧 ∩ 𝐴))) |
66 | 15 | a1i 11 |
. . . . . 6
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 = (𝑧 ∩ 𝐴)) ∧ 𝑢 ∈ 𝐿) → (𝑢 ∩ 𝐴) ∈ V) |
67 | 18 | adantr 480 |
. . . . . 6
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 = (𝑧 ∩ 𝐴)) → (𝑦 ∈ (𝐿 ↾t 𝐴) ↔ ∃𝑢 ∈ 𝐿 𝑦 = (𝑢 ∩ 𝐴))) |
68 | | ineq12 4138 |
. . . . . . . . 9
⊢ ((𝑥 = (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑢 ∩ 𝐴)) → (𝑥 ∩ 𝑦) = ((𝑧 ∩ 𝐴) ∩ (𝑢 ∩ 𝐴))) |
69 | | inindir 4158 |
. . . . . . . . 9
⊢ ((𝑧 ∩ 𝑢) ∩ 𝐴) = ((𝑧 ∩ 𝐴) ∩ (𝑢 ∩ 𝐴)) |
70 | 68, 69 | eqtr4di 2797 |
. . . . . . . 8
⊢ ((𝑥 = (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑢 ∩ 𝐴)) → (𝑥 ∩ 𝑦) = ((𝑧 ∩ 𝑢) ∩ 𝐴)) |
71 | 70 | adantll 710 |
. . . . . . 7
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 = (𝑧 ∩ 𝐴)) ∧ 𝑦 = (𝑢 ∩ 𝐴)) → (𝑥 ∩ 𝑦) = ((𝑧 ∩ 𝑢) ∩ 𝐴)) |
72 | 71 | eleq1d 2823 |
. . . . . 6
⊢ ((((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 = (𝑧 ∩ 𝐴)) ∧ 𝑦 = (𝑢 ∩ 𝐴)) → ((𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴) ↔ ((𝑧 ∩ 𝑢) ∩ 𝐴) ∈ (𝐿 ↾t 𝐴))) |
73 | 66, 67, 72 | ralxfr2d 5328 |
. . . . 5
⊢ (((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) ∧ 𝑥 = (𝑧 ∩ 𝐴)) → (∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴) ↔ ∀𝑢 ∈ 𝐿 ((𝑧 ∩ 𝑢) ∩ 𝐴) ∈ (𝐿 ↾t 𝐴))) |
74 | 63, 65, 73 | ralxfr2d 5328 |
. . . 4
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴) ↔ ∀𝑧 ∈ 𝐿 ∀𝑢 ∈ 𝐿 ((𝑧 ∩ 𝑢) ∩ 𝐴) ∈ (𝐿 ↾t 𝐴))) |
75 | 60, 74 | mpbird 256 |
. . 3
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴)) |
76 | | isfil2 22915 |
. . . . . 6
⊢ ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ (((𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ (𝐿 ↾t 𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴))) |
77 | | restsspw 17059 |
. . . . . . . 8
⊢ (𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 |
78 | | 3anass 1093 |
. . . . . . . 8
⊢ (((𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈
(𝐿 ↾t
𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ↔ ((𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 ∧ (¬ ∅ ∈ (𝐿 ↾t 𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴)))) |
79 | 77, 78 | mpbiran 705 |
. . . . . . 7
⊢ (((𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈
(𝐿 ↾t
𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ↔ (¬ ∅ ∈ (𝐿 ↾t 𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴))) |
80 | 79 | 3anbi1i 1155 |
. . . . . 6
⊢ ((((𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈
(𝐿 ↾t
𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴)) ↔ ((¬ ∅ ∈ (𝐿 ↾t 𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴))) |
81 | | 3anass 1093 |
. . . . . 6
⊢ (((¬
∅ ∈ (𝐿
↾t 𝐴)
∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴)) ↔ ((¬ ∅ ∈ (𝐿 ↾t 𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴)))) |
82 | 76, 80, 81 | 3bitri 296 |
. . . . 5
⊢ ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ((¬ ∅ ∈
(𝐿 ↾t
𝐴) ∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴)))) |
83 | | anass 468 |
. . . . 5
⊢ (((¬
∅ ∈ (𝐿
↾t 𝐴)
∧ 𝐴 ∈ (𝐿 ↾t 𝐴)) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴))) ↔ (¬ ∅ ∈ (𝐿 ↾t 𝐴) ∧ (𝐴 ∈ (𝐿 ↾t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴))))) |
84 | | ancom 460 |
. . . . 5
⊢ ((¬
∅ ∈ (𝐿
↾t 𝐴)
∧ (𝐴 ∈ (𝐿 ↾t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴)))) ↔ ((𝐴 ∈ (𝐿 ↾t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴))) ∧ ¬ ∅ ∈ (𝐿 ↾t 𝐴))) |
85 | 82, 83, 84 | 3bitri 296 |
. . . 4
⊢ ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ((𝐴 ∈ (𝐿 ↾t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴))) ∧ ¬ ∅ ∈ (𝐿 ↾t 𝐴))) |
86 | 85 | baib 535 |
. . 3
⊢ ((𝐴 ∈ (𝐿 ↾t 𝐴) ∧ (∀𝑥 ∈ 𝒫 𝐴(∃𝑦 ∈ (𝐿 ↾t 𝐴)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐿 ↾t 𝐴)) ∧ ∀𝑥 ∈ (𝐿 ↾t 𝐴)∀𝑦 ∈ (𝐿 ↾t 𝐴)(𝑥 ∩ 𝑦) ∈ (𝐿 ↾t 𝐴))) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ ∅ ∈ (𝐿 ↾t 𝐴))) |
87 | 12, 52, 75, 86 | syl12anc 833 |
. 2
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ ∅ ∈ (𝐿 ↾t 𝐴))) |
88 | | nesym 2999 |
. . . 4
⊢ ((𝑣 ∩ 𝐴) ≠ ∅ ↔ ¬ ∅ = (𝑣 ∩ 𝐴)) |
89 | 88 | ralbii 3090 |
. . 3
⊢
(∀𝑣 ∈
𝐿 (𝑣 ∩ 𝐴) ≠ ∅ ↔ ∀𝑣 ∈ 𝐿 ¬ ∅ = (𝑣 ∩ 𝐴)) |
90 | | elrest 17055 |
. . . . . 6
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V) → (∅ ∈ (𝐿 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐿 ∅ = (𝑣 ∩ 𝐴))) |
91 | 8, 90 | syldan 590 |
. . . . 5
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∅ ∈ (𝐿 ↾t 𝐴) ↔ ∃𝑣 ∈ 𝐿 ∅ = (𝑣 ∩ 𝐴))) |
92 | | dfrex2 3166 |
. . . . 5
⊢
(∃𝑣 ∈
𝐿 ∅ = (𝑣 ∩ 𝐴) ↔ ¬ ∀𝑣 ∈ 𝐿 ¬ ∅ = (𝑣 ∩ 𝐴)) |
93 | 91, 92 | bitrdi 286 |
. . . 4
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∅ ∈ (𝐿 ↾t 𝐴) ↔ ¬ ∀𝑣 ∈ 𝐿 ¬ ∅ = (𝑣 ∩ 𝐴))) |
94 | 93 | con2bid 354 |
. . 3
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∀𝑣 ∈ 𝐿 ¬ ∅ = (𝑣 ∩ 𝐴) ↔ ¬ ∅ ∈ (𝐿 ↾t 𝐴))) |
95 | 89, 94 | syl5bb 282 |
. 2
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∀𝑣 ∈ 𝐿 (𝑣 ∩ 𝐴) ≠ ∅ ↔ ¬ ∅ ∈
(𝐿 ↾t
𝐴))) |
96 | 87, 95 | bitr4d 281 |
1
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ 𝐿 (𝑣 ∩ 𝐴) ≠ ∅)) |