Step | Hyp | Ref
| Expression |
1 | | restcls.2 |
. . . . . . 7
⊢ 𝐾 = (𝐽 ↾t 𝑌) |
2 | 1 | fveq2i 6771 |
. . . . . 6
⊢
(int‘𝐾) =
(int‘(𝐽
↾t 𝑌)) |
3 | 2 | fveq1i 6769 |
. . . . 5
⊢
((int‘𝐾)‘𝑆) = ((int‘(𝐽 ↾t 𝑌))‘𝑆) |
4 | | restcls.1 |
. . . . . . . . . 10
⊢ 𝑋 = ∪
𝐽 |
5 | 4 | topopn 22036 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
6 | | ssexg 5250 |
. . . . . . . . . 10
⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V) |
7 | 6 | ancoms 458 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V) |
8 | 5, 7 | sylan 579 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V) |
9 | | resttop 22292 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽 ↾t 𝑌) ∈ Top) |
10 | 8, 9 | syldan 590 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ Top) |
11 | 10 | 3adant3 1130 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ Top) |
12 | 4 | restuni 22294 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
13 | 12 | sseq2d 3957 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝑆 ⊆ 𝑌 ↔ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌))) |
14 | 13 | biimp3a 1467 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) |
15 | | eqid 2739 |
. . . . . . 7
⊢ ∪ (𝐽
↾t 𝑌) =
∪ (𝐽 ↾t 𝑌) |
16 | 15 | ntropn 22181 |
. . . . . 6
⊢ (((𝐽 ↾t 𝑌) ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽
↾t 𝑌))
→ ((int‘(𝐽
↾t 𝑌))‘𝑆) ∈ (𝐽 ↾t 𝑌)) |
17 | 11, 14, 16 | syl2anc 583 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (𝐽 ↾t 𝑌)) |
18 | 3, 17 | eqeltrid 2844 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌)) |
19 | | simp1 1134 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top) |
20 | | uniexg 7584 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → ∪ 𝐽
∈ V) |
21 | 4, 20 | eqeltrid 2844 |
. . . . . . . 8
⊢ (𝐽 ∈ Top → 𝑋 ∈ V) |
22 | | ssexg 5250 |
. . . . . . . 8
⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝑌 ∈ V) |
23 | 21, 22 | sylan2 592 |
. . . . . . 7
⊢ ((𝑌 ⊆ 𝑋 ∧ 𝐽 ∈ Top) → 𝑌 ∈ V) |
24 | 23 | ancoms 458 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V) |
25 | 24 | 3adant3 1130 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ∈ V) |
26 | | elrest 17119 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) →
(((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) |
27 | 19, 25, 26 | syl2anc 583 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐾)‘𝑆) ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) |
28 | 18, 27 | mpbid 231 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑜 ∈ 𝐽 ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌)) |
29 | 4 | eltopss 22037 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋) |
30 | 29 | sseld 3924 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋)) |
31 | 30 | adantrr 713 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋)) |
32 | 31 | 3ad2antl1 1183 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → 𝑥 ∈ 𝑋)) |
33 | | eldif 3901 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝑋 ∖ 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝑌)) |
34 | 33 | simplbi2 500 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝑌 → 𝑥 ∈ (𝑋 ∖ 𝑌))) |
35 | 34 | orrd 859 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑋 → (𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌))) |
36 | 32, 35 | syl6 35 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → (𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)))) |
37 | | elin 3907 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝑜 ∩ 𝑌) ↔ (𝑥 ∈ 𝑜 ∧ 𝑥 ∈ 𝑌)) |
38 | | eleq2 2828 |
. . . . . . . . . . . . 13
⊢
(((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (𝑥 ∈ ((int‘𝐾)‘𝑆) ↔ 𝑥 ∈ (𝑜 ∩ 𝑌))) |
39 | | elun1 4114 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ((int‘𝐾)‘𝑆) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))) |
40 | 38, 39 | syl6bir 253 |
. . . . . . . . . . . 12
⊢
(((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (𝑥 ∈ (𝑜 ∩ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))) |
41 | 40 | ad2antll 725 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ (𝑜 ∩ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))) |
42 | 37, 41 | syl5bir 242 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → ((𝑥 ∈ 𝑜 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))) |
43 | 42 | expdimp 452 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) ∧ 𝑥 ∈ 𝑜) → (𝑥 ∈ 𝑌 → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))) |
44 | | elun2 4115 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))) |
45 | 44 | a1i 11 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) ∧ 𝑥 ∈ 𝑜) → (𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))) |
46 | 43, 45 | jaod 855 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) ∧ 𝑥 ∈ 𝑜) → ((𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))) |
47 | 46 | ex 412 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → ((𝑥 ∈ 𝑌 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))))) |
48 | 36, 47 | mpdd 43 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑥 ∈ 𝑜 → 𝑥 ∈ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)))) |
49 | 48 | ssrdv 3931 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑜 ⊆ (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌))) |
50 | 11 | adantr 480 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝐽 ↾t 𝑌) ∈ Top) |
51 | 1, 50 | eqeltrid 2844 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝐾 ∈ Top) |
52 | 14 | adantr 480 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) |
53 | 1 | unieqi 4857 |
. . . . . . . . 9
⊢ ∪ 𝐾 =
∪ (𝐽 ↾t 𝑌) |
54 | 53 | eqcomi 2748 |
. . . . . . . 8
⊢ ∪ (𝐽
↾t 𝑌) =
∪ 𝐾 |
55 | 54 | ntrss2 22189 |
. . . . . . 7
⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽
↾t 𝑌))
→ ((int‘𝐾)‘𝑆) ⊆ 𝑆) |
56 | 51, 52, 55 | syl2anc 583 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → ((int‘𝐾)‘𝑆) ⊆ 𝑆) |
57 | | unss1 4117 |
. . . . . 6
⊢
(((int‘𝐾)‘𝑆) ⊆ 𝑆 → (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌))) |
58 | 56, 57 | syl 17 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (((int‘𝐾)‘𝑆) ∪ (𝑋 ∖ 𝑌)) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌))) |
59 | 49, 58 | sstrd 3935 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌))) |
60 | | simpl1 1189 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝐽 ∈ Top) |
61 | | sstr 3933 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋) → 𝑆 ⊆ 𝑋) |
62 | 61 | ancoms 458 |
. . . . . . . . . . . . 13
⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋) |
63 | 62 | 3adant1 1128 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋) |
64 | 63 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑆 ⊆ 𝑋) |
65 | | difss 4070 |
. . . . . . . . . . 11
⊢ (𝑋 ∖ 𝑌) ⊆ 𝑋 |
66 | | unss 4122 |
. . . . . . . . . . 11
⊢ ((𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑌) ⊆ 𝑋) ↔ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) |
67 | 64, 65, 66 | sylanblc 588 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) |
68 | | simprl 767 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ∈ 𝐽) |
69 | | simprr 769 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌))) |
70 | 4 | ssntr 22190 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌)))) |
71 | 60, 67, 68, 69, 70 | syl22anc 835 |
. . . . . . . . 9
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → 𝑜 ⊆ ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌)))) |
72 | 71 | ssrind 4174 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → (𝑜 ∩ 𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)) |
73 | | sseq1 3950 |
. . . . . . . 8
⊢
(((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ↔ (𝑜 ∩ 𝑌) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))) |
74 | 72, 73 | syl5ibrcom 246 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ 𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)))) → (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))) |
75 | 74 | expr 456 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ 𝑜 ∈ 𝐽) → (𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)) → (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)))) |
76 | 75 | com23 86 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ 𝑜 ∈ 𝐽) → (((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌) → (𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)))) |
77 | 76 | impr 454 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → (𝑜 ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌)) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌))) |
78 | 59, 77 | mpd 15 |
. . 3
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑜 ∈ 𝐽 ∧ ((int‘𝐾)‘𝑆) = (𝑜 ∩ 𝑌))) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)) |
79 | 28, 78 | rexlimddv 3221 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) ⊆ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)) |
80 | 1, 11 | eqeltrid 2844 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top) |
81 | 8 | 3adant3 1130 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ∈ V) |
82 | 63, 65, 66 | sylanblc 588 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) |
83 | 4 | ntropn 22181 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∈ 𝐽) |
84 | 19, 82, 83 | syl2anc 583 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∈ 𝐽) |
85 | | elrestr 17120 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V ∧
((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∈ 𝐽) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ (𝐽 ↾t 𝑌)) |
86 | 19, 81, 84, 85 | syl3anc 1369 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ (𝐽 ↾t 𝑌)) |
87 | 86, 1 | eleqtrrdi 2851 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ 𝐾) |
88 | 4 | ntrss2 22189 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ (𝑆 ∪ (𝑋 ∖ 𝑌)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌))) |
89 | 19, 82, 88 | syl2anc 583 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ⊆ (𝑆 ∪ (𝑋 ∖ 𝑌))) |
90 | 89 | ssrind 4174 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌)) |
91 | | elin 3907 |
. . . . . . 7
⊢ (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ↔ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ∧ 𝑥 ∈ 𝑌)) |
92 | | elun 4087 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ↔ (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌))) |
93 | | orcom 866 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) ↔ (𝑥 ∈ (𝑋 ∖ 𝑌) ∨ 𝑥 ∈ 𝑆)) |
94 | | df-or 844 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (𝑋 ∖ 𝑌) ∨ 𝑥 ∈ 𝑆) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆)) |
95 | 93, 94 | bitri 274 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑆 ∨ 𝑥 ∈ (𝑋 ∖ 𝑌)) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆)) |
96 | 92, 95 | bitri 274 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ↔ (¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆)) |
97 | 96 | anbi1i 623 |
. . . . . . 7
⊢ ((𝑥 ∈ (𝑆 ∪ (𝑋 ∖ 𝑌)) ∧ 𝑥 ∈ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌)) |
98 | 91, 97 | bitri 274 |
. . . . . 6
⊢ (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ↔ ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌)) |
99 | | elndif 4067 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑌 → ¬ 𝑥 ∈ (𝑋 ∖ 𝑌)) |
100 | | pm2.27 42 |
. . . . . . . . 9
⊢ (¬
𝑥 ∈ (𝑋 ∖ 𝑌) → ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)) |
101 | 99, 100 | syl 17 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑌 → ((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)) |
102 | 101 | impcom 407 |
. . . . . . 7
⊢ (((¬
𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑆) |
103 | 102 | a1i 11 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((¬ 𝑥 ∈ (𝑋 ∖ 𝑌) → 𝑥 ∈ 𝑆) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑆)) |
104 | 98, 103 | syl5bi 241 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) → 𝑥 ∈ 𝑆)) |
105 | 104 | ssrdv 3931 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((𝑆 ∪ (𝑋 ∖ 𝑌)) ∩ 𝑌) ⊆ 𝑆) |
106 | 90, 105 | sstrd 3935 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ 𝑆) |
107 | 54 | ssntr 22190 |
. . 3
⊢ (((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽
↾t 𝑌))
∧ ((((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ∈ 𝐾 ∧ (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ 𝑆)) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆)) |
108 | 80, 14, 87, 106, 107 | syl22anc 835 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌) ⊆ ((int‘𝐾)‘𝑆)) |
109 | 79, 108 | eqssd 3942 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((int‘𝐾)‘𝑆) = (((int‘𝐽)‘(𝑆 ∪ (𝑋 ∖ 𝑌))) ∩ 𝑌)) |