Step | Hyp | Ref
| Expression |
1 | | id 22 |
. . . . 5
⊢ (𝑆 ⊆ 𝑋 → 𝑆 ⊆ 𝑋) |
2 | | restcld.1 |
. . . . . 6
⊢ 𝑋 = ∪
𝐽 |
3 | 2 | topopn 20711 |
. . . . 5
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
4 | | ssexg 4804 |
. . . . 5
⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑆 ∈ V) |
5 | 1, 3, 4 | syl2anr 495 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V) |
6 | | resttop 20964 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾t 𝑆) ∈ Top) |
7 | 5, 6 | syldan 487 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ↾t 𝑆) ∈ Top) |
8 | | eqid 2622 |
. . . 4
⊢ ∪ (𝐽
↾t 𝑆) =
∪ (𝐽 ↾t 𝑆) |
9 | 8 | iscld 20831 |
. . 3
⊢ ((𝐽 ↾t 𝑆) ∈ Top → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ ∪ (𝐽 ↾t 𝑆) ∧ (∪ (𝐽
↾t 𝑆)
∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))) |
10 | 7, 9 | syl 17 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ ∪ (𝐽 ↾t 𝑆) ∧ (∪ (𝐽
↾t 𝑆)
∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))) |
11 | 2 | restuni 20966 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾t 𝑆)) |
12 | 11 | sseq2d 3633 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ⊆ 𝑆 ↔ 𝐴 ⊆ ∪ (𝐽 ↾t 𝑆))) |
13 | 11 | difeq1d 3727 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝐴) = (∪ (𝐽 ↾t 𝑆) ∖ 𝐴)) |
14 | 13 | eleq1d 2686 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (∪
(𝐽 ↾t
𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))) |
15 | 12, 14 | anbi12d 747 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ ∪ (𝐽 ↾t 𝑆) ∧ (∪ (𝐽
↾t 𝑆)
∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))) |
16 | | elrest 16088 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆))) |
17 | 5, 16 | syldan 487 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆))) |
18 | 17 | anbi2d 740 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))) |
19 | 2 | opncld 20837 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽)) |
20 | 19 | adantlr 751 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽)) |
21 | 20 | adantlr 751 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽)) |
22 | 21 | adantr 481 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽)) |
23 | | incom 3805 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∩ 𝑆) = (𝑆 ∩ 𝑋) |
24 | | df-ss 3588 |
. . . . . . . . . . . . . 14
⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆) |
25 | 24 | biimpi 206 |
. . . . . . . . . . . . 13
⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∩ 𝑋) = 𝑆) |
26 | 23, 25 | syl5eq 2668 |
. . . . . . . . . . . 12
⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∩ 𝑆) = 𝑆) |
27 | 26 | ad4antlr 769 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈ Top
∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∩ 𝑆) = 𝑆) |
28 | 27 | difeq1d 3727 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈ Top
∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ((𝑋 ∩ 𝑆) ∖ 𝑜) = (𝑆 ∖ 𝑜)) |
29 | | difeq2 3722 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ (𝑜 ∩ 𝑆))) |
30 | | difindi 3881 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆)) |
31 | | difid 3948 |
. . . . . . . . . . . . . 14
⊢ (𝑆 ∖ 𝑆) = ∅ |
32 | 31 | uneq2i 3764 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ ∅) |
33 | | un0 3967 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∖ 𝑜) ∪ ∅) = (𝑆 ∖ 𝑜) |
34 | 30, 32, 33 | 3eqtri 2648 |
. . . . . . . . . . . 12
⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = (𝑆 ∖ 𝑜) |
35 | 29, 34 | syl6eq 2672 |
. . . . . . . . . . 11
⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜)) |
36 | 35 | adantl 482 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈ Top
∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜)) |
37 | | dfss4 3858 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴) |
38 | 37 | biimpi 206 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ 𝑆 → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴) |
39 | 38 | ad3antlr 767 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈ Top
∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴) |
40 | 28, 36, 39 | 3eqtr2rd 2663 |
. . . . . . . . 9
⊢
(((((𝐽 ∈ Top
∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∩ 𝑆) ∖ 𝑜)) |
41 | 23 | difeq1i 3724 |
. . . . . . . . . 10
⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑆 ∩ 𝑋) ∖ 𝑜) |
42 | | indif2 3870 |
. . . . . . . . . 10
⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑆 ∩ 𝑋) ∖ 𝑜) |
43 | | incom 3805 |
. . . . . . . . . 10
⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑋 ∖ 𝑜) ∩ 𝑆) |
44 | 41, 42, 43 | 3eqtr2i 2650 |
. . . . . . . . 9
⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑋 ∖ 𝑜) ∩ 𝑆) |
45 | 40, 44 | syl6eq 2672 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆)) |
46 | | ineq1 3807 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑋 ∖ 𝑜) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ 𝑜) ∩ 𝑆)) |
47 | 46 | eqeq2d 2632 |
. . . . . . . . 9
⊢ (𝑥 = (𝑋 ∖ 𝑜) → (𝐴 = (𝑥 ∩ 𝑆) ↔ 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆))) |
48 | 47 | rspcev 3309 |
. . . . . . . 8
⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)) |
49 | 22, 45, 48 | syl2anc 693 |
. . . . . . 7
⊢
(((((𝐽 ∈ Top
∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)) |
50 | 49 | ex 450 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) → ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))) |
51 | 50 | rexlimdva 3031 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) → (∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))) |
52 | 51 | expimpd 629 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))) |
53 | 18, 52 | sylbid 230 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))) |
54 | | difindi 3881 |
. . . . . . . . . 10
⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆)) |
55 | 31 | uneq2i 3764 |
. . . . . . . . . 10
⊢ ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ ∅) |
56 | | un0 3967 |
. . . . . . . . . 10
⊢ ((𝑆 ∖ 𝑥) ∪ ∅) = (𝑆 ∖ 𝑥) |
57 | 54, 55, 56 | 3eqtri 2648 |
. . . . . . . . 9
⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = (𝑆 ∖ 𝑥) |
58 | | difin2 3890 |
. . . . . . . . . 10
⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆)) |
59 | 58 | adantl 482 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆)) |
60 | 57, 59 | syl5eq 2668 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆)) |
61 | 60 | adantr 481 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆)) |
62 | | simpll 790 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top) |
63 | 5 | adantr 481 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V) |
64 | 2 | cldopn 20835 |
. . . . . . . . 9
⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽) |
65 | 64 | adantl 482 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽) |
66 | | elrestr 16089 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋 ∖ 𝑥) ∈ 𝐽) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) |
67 | 62, 63, 65, 66 | syl3anc 1326 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆)) |
68 | 61, 67 | eqeltrd 2701 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)) |
69 | | inss2 3834 |
. . . . . 6
⊢ (𝑥 ∩ 𝑆) ⊆ 𝑆 |
70 | 68, 69 | jctil 560 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))) |
71 | | sseq1 3626 |
. . . . . 6
⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ↔ (𝑥 ∩ 𝑆) ⊆ 𝑆)) |
72 | | difeq2 3722 |
. . . . . . 7
⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝑆 ∖ 𝐴) = (𝑆 ∖ (𝑥 ∩ 𝑆))) |
73 | 72 | eleq1d 2686 |
. . . . . 6
⊢ (𝐴 = (𝑥 ∩ 𝑆) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))) |
74 | 71, 73 | anbi12d 747 |
. . . . 5
⊢ (𝐴 = (𝑥 ∩ 𝑆) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))) |
75 | 70, 74 | syl5ibrcom 237 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))) |
76 | 75 | rexlimdva 3031 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))) |
77 | 53, 76 | impbid 202 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))) |
78 | 10, 15, 77 | 3bitr2d 296 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))) |