Step | Hyp | Ref
| Expression |
1 | | unieq 4804 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → ∪ 𝑎 =
∪ ∅) |
2 | | uni0 4823 |
. . . . . . . . 9
⊢ ∪ ∅ = ∅ |
3 | 1, 2 | eqtrdi 2789 |
. . . . . . . 8
⊢ (𝑎 = ∅ → ∪ 𝑎 =
∅) |
4 | 3 | eleq1d 2817 |
. . . . . . 7
⊢ (𝑎 = ∅ → (∪ 𝑎
∈ 𝐽 ↔ ∅
∈ 𝐽)) |
5 | | mretopd.j |
. . . . . . . . . . . . . 14
⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} |
6 | 5 | ssrab3 3969 |
. . . . . . . . . . . . 13
⊢ 𝐽 ⊆ 𝒫 𝐵 |
7 | | sstr 3883 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝐵) → 𝑎 ⊆ 𝒫 𝐵) |
8 | 6, 7 | mpan2 691 |
. . . . . . . . . . . 12
⊢ (𝑎 ⊆ 𝐽 → 𝑎 ⊆ 𝒫 𝐵) |
9 | 8 | adantl 485 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → 𝑎 ⊆ 𝒫 𝐵) |
10 | | sspwuni 4982 |
. . . . . . . . . . 11
⊢ (𝑎 ⊆ 𝒫 𝐵 ↔ ∪ 𝑎
⊆ 𝐵) |
11 | 9, 10 | sylib 221 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∪ 𝑎 ⊆ 𝐵) |
12 | | vuniex 7477 |
. . . . . . . . . . 11
⊢ ∪ 𝑎
∈ V |
13 | 12 | elpw 4489 |
. . . . . . . . . 10
⊢ (∪ 𝑎
∈ 𝒫 𝐵 ↔
∪ 𝑎 ⊆ 𝐵) |
14 | 11, 13 | sylibr 237 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∪ 𝑎 ∈ 𝒫 𝐵) |
15 | 14 | adantr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∪ 𝑎
∈ 𝒫 𝐵) |
16 | | uniiun 4941 |
. . . . . . . . . 10
⊢ ∪ 𝑎 =
∪ 𝑏 ∈ 𝑎 𝑏 |
17 | 16 | difeq2i 4008 |
. . . . . . . . 9
⊢ (𝐵 ∖ ∪ 𝑎) =
(𝐵 ∖ ∪ 𝑏 ∈ 𝑎 𝑏) |
18 | | iindif2 4959 |
. . . . . . . . . . 11
⊢ (𝑎 ≠ ∅ → ∩ 𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) = (𝐵 ∖ ∪
𝑏 ∈ 𝑎 𝑏)) |
19 | 18 | adantl 485 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∩ 𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) = (𝐵 ∖ ∪
𝑏 ∈ 𝑎 𝑏)) |
20 | | mretopd.m |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ (Moore‘𝐵)) |
21 | 20 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → 𝑀 ∈ (Moore‘𝐵)) |
22 | | simpr 488 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅) |
23 | | difeq2 4005 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑏 → (𝐵 ∖ 𝑧) = (𝐵 ∖ 𝑏)) |
24 | 23 | eleq1d 2817 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑏 → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ 𝑏) ∈ 𝑀)) |
25 | 24, 5 | elrab2 3588 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ 𝐽 ↔ (𝑏 ∈ 𝒫 𝐵 ∧ (𝐵 ∖ 𝑏) ∈ 𝑀)) |
26 | 25 | simprbi 500 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ 𝐽 → (𝐵 ∖ 𝑏) ∈ 𝑀) |
27 | 26 | rgen 3063 |
. . . . . . . . . . . . 13
⊢
∀𝑏 ∈
𝐽 (𝐵 ∖ 𝑏) ∈ 𝑀 |
28 | | ssralv 3941 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ⊆ 𝐽 → (∀𝑏 ∈ 𝐽 (𝐵 ∖ 𝑏) ∈ 𝑀 → ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀)) |
29 | 28 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → (∀𝑏 ∈ 𝐽 (𝐵 ∖ 𝑏) ∈ 𝑀 → ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀)) |
30 | 27, 29 | mpi 20 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) |
31 | 30 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) |
32 | | mreiincl 16963 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ (Moore‘𝐵) ∧ 𝑎 ≠ ∅ ∧ ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) → ∩
𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) |
33 | 21, 22, 31, 32 | syl3anc 1372 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∩ 𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) |
34 | 19, 33 | eqeltrrd 2834 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → (𝐵 ∖ ∪
𝑏 ∈ 𝑎 𝑏) ∈ 𝑀) |
35 | 17, 34 | eqeltrid 2837 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → (𝐵 ∖ ∪ 𝑎) ∈ 𝑀) |
36 | | difeq2 4005 |
. . . . . . . . . 10
⊢ (𝑧 = ∪
𝑎 → (𝐵 ∖ 𝑧) = (𝐵 ∖ ∪ 𝑎)) |
37 | 36 | eleq1d 2817 |
. . . . . . . . 9
⊢ (𝑧 = ∪
𝑎 → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ ∪ 𝑎) ∈ 𝑀)) |
38 | 37, 5 | elrab2 3588 |
. . . . . . . 8
⊢ (∪ 𝑎
∈ 𝐽 ↔ (∪ 𝑎
∈ 𝒫 𝐵 ∧
(𝐵 ∖ ∪ 𝑎)
∈ 𝑀)) |
39 | 15, 35, 38 | sylanbrc 586 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∪ 𝑎
∈ 𝐽) |
40 | | 0elpw 5219 |
. . . . . . . . . 10
⊢ ∅
∈ 𝒫 𝐵 |
41 | 40 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ∅ ∈ 𝒫
𝐵) |
42 | | mre1cl 16961 |
. . . . . . . . . 10
⊢ (𝑀 ∈ (Moore‘𝐵) → 𝐵 ∈ 𝑀) |
43 | 20, 42 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ 𝑀) |
44 | | difeq2 4005 |
. . . . . . . . . . . 12
⊢ (𝑧 = ∅ → (𝐵 ∖ 𝑧) = (𝐵 ∖ ∅)) |
45 | | dif0 4259 |
. . . . . . . . . . . 12
⊢ (𝐵 ∖ ∅) = 𝐵 |
46 | 44, 45 | eqtrdi 2789 |
. . . . . . . . . . 11
⊢ (𝑧 = ∅ → (𝐵 ∖ 𝑧) = 𝐵) |
47 | 46 | eleq1d 2817 |
. . . . . . . . . 10
⊢ (𝑧 = ∅ → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ 𝐵 ∈ 𝑀)) |
48 | 47, 5 | elrab2 3588 |
. . . . . . . . 9
⊢ (∅
∈ 𝐽 ↔ (∅
∈ 𝒫 𝐵 ∧
𝐵 ∈ 𝑀)) |
49 | 41, 43, 48 | sylanbrc 586 |
. . . . . . . 8
⊢ (𝜑 → ∅ ∈ 𝐽) |
50 | 49 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∅ ∈ 𝐽) |
51 | 4, 39, 50 | pm2.61ne 3019 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∪ 𝑎 ∈ 𝐽) |
52 | 51 | ex 416 |
. . . . 5
⊢ (𝜑 → (𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽)) |
53 | 52 | alrimiv 1933 |
. . . 4
⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽)) |
54 | | inss1 4117 |
. . . . . . . 8
⊢ (𝑎 ∩ 𝑏) ⊆ 𝑎 |
55 | | difeq2 4005 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑎 → (𝐵 ∖ 𝑧) = (𝐵 ∖ 𝑎)) |
56 | 55 | eleq1d 2817 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑎 → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ 𝑎) ∈ 𝑀)) |
57 | 56, 5 | elrab2 3588 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ 𝐽 ↔ (𝑎 ∈ 𝒫 𝐵 ∧ (𝐵 ∖ 𝑎) ∈ 𝑀)) |
58 | 57 | simplbi 501 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝐽 → 𝑎 ∈ 𝒫 𝐵) |
59 | 58 | elpwid 4496 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐽 → 𝑎 ⊆ 𝐵) |
60 | 59 | ad2antrl 728 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → 𝑎 ⊆ 𝐵) |
61 | 54, 60 | sstrid 3886 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝑎 ∩ 𝑏) ⊆ 𝐵) |
62 | | vex 3401 |
. . . . . . . . 9
⊢ 𝑎 ∈ V |
63 | 62 | inex1 5182 |
. . . . . . . 8
⊢ (𝑎 ∩ 𝑏) ∈ V |
64 | 63 | elpw 4489 |
. . . . . . 7
⊢ ((𝑎 ∩ 𝑏) ∈ 𝒫 𝐵 ↔ (𝑎 ∩ 𝑏) ⊆ 𝐵) |
65 | 61, 64 | sylibr 237 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝑎 ∩ 𝑏) ∈ 𝒫 𝐵) |
66 | | difindi 4170 |
. . . . . . 7
⊢ (𝐵 ∖ (𝑎 ∩ 𝑏)) = ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) |
67 | 57 | simprbi 500 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐽 → (𝐵 ∖ 𝑎) ∈ 𝑀) |
68 | 67 | ad2antrl 728 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝐵 ∖ 𝑎) ∈ 𝑀) |
69 | 26 | ad2antll 729 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝐵 ∖ 𝑏) ∈ 𝑀) |
70 | | simpl 486 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → 𝜑) |
71 | | uneq1 4044 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐵 ∖ 𝑎) → (𝑥 ∪ 𝑦) = ((𝐵 ∖ 𝑎) ∪ 𝑦)) |
72 | 71 | eleq1d 2817 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝐵 ∖ 𝑎) → ((𝑥 ∪ 𝑦) ∈ 𝑀 ↔ ((𝐵 ∖ 𝑎) ∪ 𝑦) ∈ 𝑀)) |
73 | 72 | imbi2d 344 |
. . . . . . . . . 10
⊢ (𝑥 = (𝐵 ∖ 𝑎) → ((𝜑 → (𝑥 ∪ 𝑦) ∈ 𝑀) ↔ (𝜑 → ((𝐵 ∖ 𝑎) ∪ 𝑦) ∈ 𝑀))) |
74 | | uneq2 4045 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐵 ∖ 𝑏) → ((𝐵 ∖ 𝑎) ∪ 𝑦) = ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏))) |
75 | 74 | eleq1d 2817 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐵 ∖ 𝑏) → (((𝐵 ∖ 𝑎) ∪ 𝑦) ∈ 𝑀 ↔ ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀)) |
76 | 75 | imbi2d 344 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐵 ∖ 𝑏) → ((𝜑 → ((𝐵 ∖ 𝑎) ∪ 𝑦) ∈ 𝑀) ↔ (𝜑 → ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀))) |
77 | | mretopd.u |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀) → (𝑥 ∪ 𝑦) ∈ 𝑀) |
78 | 77 | 3expb 1121 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) → (𝑥 ∪ 𝑦) ∈ 𝑀) |
79 | 78 | expcom 417 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀) → (𝜑 → (𝑥 ∪ 𝑦) ∈ 𝑀)) |
80 | 73, 76, 79 | vtocl2ga 3478 |
. . . . . . . . 9
⊢ (((𝐵 ∖ 𝑎) ∈ 𝑀 ∧ (𝐵 ∖ 𝑏) ∈ 𝑀) → (𝜑 → ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀)) |
81 | 80 | imp 410 |
. . . . . . . 8
⊢ ((((𝐵 ∖ 𝑎) ∈ 𝑀 ∧ (𝐵 ∖ 𝑏) ∈ 𝑀) ∧ 𝜑) → ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀) |
82 | 68, 69, 70, 81 | syl21anc 837 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀) |
83 | 66, 82 | eqeltrid 2837 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝐵 ∖ (𝑎 ∩ 𝑏)) ∈ 𝑀) |
84 | | difeq2 4005 |
. . . . . . . 8
⊢ (𝑧 = (𝑎 ∩ 𝑏) → (𝐵 ∖ 𝑧) = (𝐵 ∖ (𝑎 ∩ 𝑏))) |
85 | 84 | eleq1d 2817 |
. . . . . . 7
⊢ (𝑧 = (𝑎 ∩ 𝑏) → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ (𝑎 ∩ 𝑏)) ∈ 𝑀)) |
86 | 85, 5 | elrab2 3588 |
. . . . . 6
⊢ ((𝑎 ∩ 𝑏) ∈ 𝐽 ↔ ((𝑎 ∩ 𝑏) ∈ 𝒫 𝐵 ∧ (𝐵 ∖ (𝑎 ∩ 𝑏)) ∈ 𝑀)) |
87 | 65, 83, 86 | sylanbrc 586 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝑎 ∩ 𝑏) ∈ 𝐽) |
88 | 87 | ralrimivva 3103 |
. . . 4
⊢ (𝜑 → ∀𝑎 ∈ 𝐽 ∀𝑏 ∈ 𝐽 (𝑎 ∩ 𝑏) ∈ 𝐽) |
89 | 43 | pwexd 5243 |
. . . . . 6
⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
90 | 5, 89 | rabexd 5198 |
. . . . 5
⊢ (𝜑 → 𝐽 ∈ V) |
91 | | istopg 21639 |
. . . . 5
⊢ (𝐽 ∈ V → (𝐽 ∈ Top ↔
(∀𝑎(𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽) ∧ ∀𝑎 ∈ 𝐽 ∀𝑏 ∈ 𝐽 (𝑎 ∩ 𝑏) ∈ 𝐽))) |
92 | 90, 91 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐽 ∈ Top ↔ (∀𝑎(𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽) ∧ ∀𝑎 ∈ 𝐽 ∀𝑏 ∈ 𝐽 (𝑎 ∩ 𝑏) ∈ 𝐽))) |
93 | 53, 88, 92 | mpbir2and 713 |
. . 3
⊢ (𝜑 → 𝐽 ∈ Top) |
94 | 6 | unissi 4802 |
. . . . . 6
⊢ ∪ 𝐽
⊆ ∪ 𝒫 𝐵 |
95 | | unipw 5306 |
. . . . . 6
⊢ ∪ 𝒫 𝐵 = 𝐵 |
96 | 94, 95 | sseqtri 3911 |
. . . . 5
⊢ ∪ 𝐽
⊆ 𝐵 |
97 | | pwidg 4507 |
. . . . . . 7
⊢ (𝐵 ∈ 𝑀 → 𝐵 ∈ 𝒫 𝐵) |
98 | 43, 97 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵) |
99 | | difid 4257 |
. . . . . . 7
⊢ (𝐵 ∖ 𝐵) = ∅ |
100 | | mretopd.z |
. . . . . . 7
⊢ (𝜑 → ∅ ∈ 𝑀) |
101 | 99, 100 | eqeltrid 2837 |
. . . . . 6
⊢ (𝜑 → (𝐵 ∖ 𝐵) ∈ 𝑀) |
102 | | difeq2 4005 |
. . . . . . . 8
⊢ (𝑧 = 𝐵 → (𝐵 ∖ 𝑧) = (𝐵 ∖ 𝐵)) |
103 | 102 | eleq1d 2817 |
. . . . . . 7
⊢ (𝑧 = 𝐵 → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ 𝐵) ∈ 𝑀)) |
104 | 103, 5 | elrab2 3588 |
. . . . . 6
⊢ (𝐵 ∈ 𝐽 ↔ (𝐵 ∈ 𝒫 𝐵 ∧ (𝐵 ∖ 𝐵) ∈ 𝑀)) |
105 | 98, 101, 104 | sylanbrc 586 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ 𝐽) |
106 | | unissel 4826 |
. . . . 5
⊢ ((∪ 𝐽
⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ∪ 𝐽 = 𝐵) |
107 | 96, 105, 106 | sylancr 590 |
. . . 4
⊢ (𝜑 → ∪ 𝐽 =
𝐵) |
108 | 107 | eqcomd 2744 |
. . 3
⊢ (𝜑 → 𝐵 = ∪ 𝐽) |
109 | | istopon 21656 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
110 | 93, 108, 109 | sylanbrc 586 |
. 2
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵)) |
111 | | eqid 2738 |
. . . . 5
⊢ ∪ 𝐽 =
∪ 𝐽 |
112 | 111 | cldval 21767 |
. . . 4
⊢ (𝐽 ∈ Top →
(Clsd‘𝐽) = {𝑥 ∈ 𝒫 ∪ 𝐽
∣ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽}) |
113 | 93, 112 | syl 17 |
. . 3
⊢ (𝜑 → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 ∪ 𝐽
∣ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽}) |
114 | 107 | pweqd 4504 |
. . . 4
⊢ (𝜑 → 𝒫 ∪ 𝐽 =
𝒫 𝐵) |
115 | 107 | difeq1d 4010 |
. . . . 5
⊢ (𝜑 → (∪ 𝐽
∖ 𝑥) = (𝐵 ∖ 𝑥)) |
116 | 115 | eleq1d 2817 |
. . . 4
⊢ (𝜑 → ((∪ 𝐽
∖ 𝑥) ∈ 𝐽 ↔ (𝐵 ∖ 𝑥) ∈ 𝐽)) |
117 | 114, 116 | rabeqbidv 3386 |
. . 3
⊢ (𝜑 → {𝑥 ∈ 𝒫 ∪ 𝐽
∣ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽} = {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑥) ∈ 𝐽}) |
118 | 5 | eleq2i 2824 |
. . . . . . 7
⊢ ((𝐵 ∖ 𝑥) ∈ 𝐽 ↔ (𝐵 ∖ 𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀}) |
119 | | difss 4020 |
. . . . . . . . . 10
⊢ (𝐵 ∖ 𝑥) ⊆ 𝐵 |
120 | | elpw2g 5209 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ 𝑀 → ((𝐵 ∖ 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∖ 𝑥) ⊆ 𝐵)) |
121 | 43, 120 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐵 ∖ 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∖ 𝑥) ⊆ 𝐵)) |
122 | 119, 121 | mpbiri 261 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵 ∖ 𝑥) ∈ 𝒫 𝐵) |
123 | | difeq2 4005 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝐵 ∖ 𝑥) → (𝐵 ∖ 𝑧) = (𝐵 ∖ (𝐵 ∖ 𝑥))) |
124 | 123 | eleq1d 2817 |
. . . . . . . . . 10
⊢ (𝑧 = (𝐵 ∖ 𝑥) → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
125 | 124 | elrab3 3586 |
. . . . . . . . 9
⊢ ((𝐵 ∖ 𝑥) ∈ 𝒫 𝐵 → ((𝐵 ∖ 𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
126 | 122, 125 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵 ∖ 𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
127 | 126 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
128 | 118, 127 | syl5bb 286 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑥) ∈ 𝐽 ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
129 | | elpwi 4494 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵) |
130 | | dfss4 4147 |
. . . . . . . . 9
⊢ (𝑥 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) = 𝑥) |
131 | 129, 130 | sylib 221 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑥)) = 𝑥) |
132 | 131 | adantl 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑥)) = 𝑥) |
133 | 132 | eleq1d 2817 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀 ↔ 𝑥 ∈ 𝑀)) |
134 | 128, 133 | bitrd 282 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝑀)) |
135 | 134 | rabbidva 3378 |
. . . 4
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑥) ∈ 𝐽} = {𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀}) |
136 | | incom 4089 |
. . . . . 6
⊢ (𝑀 ∩ 𝒫 𝐵) = (𝒫 𝐵 ∩ 𝑀) |
137 | | dfin5 3849 |
. . . . . 6
⊢
(𝒫 𝐵 ∩
𝑀) = {𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀} |
138 | 136, 137 | eqtri 2761 |
. . . . 5
⊢ (𝑀 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀} |
139 | | mresspw 16959 |
. . . . . . 7
⊢ (𝑀 ∈ (Moore‘𝐵) → 𝑀 ⊆ 𝒫 𝐵) |
140 | 20, 139 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ⊆ 𝒫 𝐵) |
141 | | df-ss 3858 |
. . . . . 6
⊢ (𝑀 ⊆ 𝒫 𝐵 ↔ (𝑀 ∩ 𝒫 𝐵) = 𝑀) |
142 | 140, 141 | sylib 221 |
. . . . 5
⊢ (𝜑 → (𝑀 ∩ 𝒫 𝐵) = 𝑀) |
143 | 138, 142 | eqtr3id 2787 |
. . . 4
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀} = 𝑀) |
144 | 135, 143 | eqtrd 2773 |
. . 3
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑥) ∈ 𝐽} = 𝑀) |
145 | 113, 117,
144 | 3eqtrrd 2778 |
. 2
⊢ (𝜑 → 𝑀 = (Clsd‘𝐽)) |
146 | 110, 145 | jca 515 |
1
⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ 𝑀 = (Clsd‘𝐽))) |