Step | Hyp | Ref
| Expression |
1 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑑(𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) |
2 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑑 𝑐 ⊆ ∪ (𝐽
↾t 𝐴) |
3 | | nfre1 3309 |
. . . . . . 7
⊢
Ⅎ𝑑∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐) |
4 | 2, 3 | nfan 1899 |
. . . . . 6
⊢
Ⅎ𝑑(𝑐 ⊆ ∪ (𝐽
↾t 𝐴)
∧ ∃𝑑 ∈
(𝐽 ↾t
𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) |
5 | 1, 4 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑑((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐))) |
6 | | simpl 485 |
. . . . . . 7
⊢ ((𝑐 ⊆ ∪ (𝐽
↾t 𝐴)
∧ ∃𝑑 ∈
(𝐽 ↾t
𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) |
7 | 6 | anim2i 618 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐))) → ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴))) |
8 | | simp-5r 784 |
. . . . . . . . . . 11
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) |
9 | | simp1 1132 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top) |
10 | | simp2 1133 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝑋) |
11 | | neitr.1 |
. . . . . . . . . . . . . 14
⊢ 𝑋 = ∪
𝐽 |
12 | 11 | restuni 21773 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
13 | 9, 10, 12 | syl2anc 586 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
14 | 13 | ad5antr 732 |
. . . . . . . . . . 11
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
15 | 8, 14 | sseqtrrd 4011 |
. . . . . . . . . 10
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑐 ⊆ 𝐴) |
16 | 10 | ad5antr 732 |
. . . . . . . . . 10
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝐴 ⊆ 𝑋) |
17 | 15, 16 | sstrd 3980 |
. . . . . . . . 9
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑐 ⊆ 𝑋) |
18 | 9 | ad5antr 732 |
. . . . . . . . . . 11
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝐽 ∈ Top) |
19 | | simplr 767 |
. . . . . . . . . . 11
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑒 ∈ 𝐽) |
20 | 11 | eltopss 21518 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑒 ∈ 𝐽) → 𝑒 ⊆ 𝑋) |
21 | 18, 19, 20 | syl2anc 586 |
. . . . . . . . . 10
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑒 ⊆ 𝑋) |
22 | 21 | ssdifssd 4122 |
. . . . . . . . 9
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → (𝑒 ∖ 𝐴) ⊆ 𝑋) |
23 | 17, 22 | unssd 4165 |
. . . . . . . 8
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → (𝑐 ∪ (𝑒 ∖ 𝐴)) ⊆ 𝑋) |
24 | | simpr1l 1226 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ ((𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐) ∧ 𝑒 ∈ 𝐽 ∧ 𝑑 = (𝑒 ∩ 𝐴))) → 𝐵 ⊆ 𝑑) |
25 | 24 | 3anassrs 1356 |
. . . . . . . . . . 11
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝐵 ⊆ 𝑑) |
26 | | simpr 487 |
. . . . . . . . . . 11
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑑 = (𝑒 ∩ 𝐴)) |
27 | 25, 26 | sseqtrd 4010 |
. . . . . . . . . 10
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝐵 ⊆ (𝑒 ∩ 𝐴)) |
28 | | inss1 4208 |
. . . . . . . . . 10
⊢ (𝑒 ∩ 𝐴) ⊆ 𝑒 |
29 | 27, 28 | sstrdi 3982 |
. . . . . . . . 9
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝐵 ⊆ 𝑒) |
30 | | inundif 4430 |
. . . . . . . . . 10
⊢ ((𝑒 ∩ 𝐴) ∪ (𝑒 ∖ 𝐴)) = 𝑒 |
31 | | simpr1r 1227 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ ((𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐) ∧ 𝑒 ∈ 𝐽 ∧ 𝑑 = (𝑒 ∩ 𝐴))) → 𝑑 ⊆ 𝑐) |
32 | 31 | 3anassrs 1356 |
. . . . . . . . . . . 12
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑑 ⊆ 𝑐) |
33 | 26, 32 | eqsstrrd 4009 |
. . . . . . . . . . 11
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → (𝑒 ∩ 𝐴) ⊆ 𝑐) |
34 | | unss1 4158 |
. . . . . . . . . . 11
⊢ ((𝑒 ∩ 𝐴) ⊆ 𝑐 → ((𝑒 ∩ 𝐴) ∪ (𝑒 ∖ 𝐴)) ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))) |
35 | 33, 34 | syl 17 |
. . . . . . . . . 10
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → ((𝑒 ∩ 𝐴) ∪ (𝑒 ∖ 𝐴)) ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))) |
36 | 30, 35 | eqsstrrid 4019 |
. . . . . . . . 9
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑒 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))) |
37 | | sseq2 3996 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑒 → (𝐵 ⊆ 𝑏 ↔ 𝐵 ⊆ 𝑒)) |
38 | | sseq1 3995 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑒 → (𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)) ↔ 𝑒 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) |
39 | 37, 38 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑒 → ((𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))) ↔ (𝐵 ⊆ 𝑒 ∧ 𝑒 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))))) |
40 | 39 | rspcev 3626 |
. . . . . . . . 9
⊢ ((𝑒 ∈ 𝐽 ∧ (𝐵 ⊆ 𝑒 ∧ 𝑒 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) → ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) |
41 | 19, 29, 36, 40 | syl12anc 834 |
. . . . . . . 8
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) |
42 | | indir 4255 |
. . . . . . . . . . 11
⊢ ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴) = ((𝑐 ∩ 𝐴) ∪ ((𝑒 ∖ 𝐴) ∩ 𝐴)) |
43 | | incom 4181 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∩ (𝑒 ∖ 𝐴)) = ((𝑒 ∖ 𝐴) ∩ 𝐴) |
44 | | disjdif 4424 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∩ (𝑒 ∖ 𝐴)) = ∅ |
45 | 43, 44 | eqtr3i 2849 |
. . . . . . . . . . . 12
⊢ ((𝑒 ∖ 𝐴) ∩ 𝐴) = ∅ |
46 | 45 | uneq2i 4139 |
. . . . . . . . . . 11
⊢ ((𝑐 ∩ 𝐴) ∪ ((𝑒 ∖ 𝐴) ∩ 𝐴)) = ((𝑐 ∩ 𝐴) ∪ ∅) |
47 | | un0 4347 |
. . . . . . . . . . 11
⊢ ((𝑐 ∩ 𝐴) ∪ ∅) = (𝑐 ∩ 𝐴) |
48 | 42, 46, 47 | 3eqtri 2851 |
. . . . . . . . . 10
⊢ ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴) = (𝑐 ∩ 𝐴) |
49 | | df-ss 3955 |
. . . . . . . . . . 11
⊢ (𝑐 ⊆ 𝐴 ↔ (𝑐 ∩ 𝐴) = 𝑐) |
50 | 49 | biimpi 218 |
. . . . . . . . . 10
⊢ (𝑐 ⊆ 𝐴 → (𝑐 ∩ 𝐴) = 𝑐) |
51 | 48, 50 | syl5req 2872 |
. . . . . . . . 9
⊢ (𝑐 ⊆ 𝐴 → 𝑐 = ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴)) |
52 | 15, 51 | syl 17 |
. . . . . . . 8
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → 𝑐 = ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴)) |
53 | | vex 3500 |
. . . . . . . . . 10
⊢ 𝑐 ∈ V |
54 | | vex 3500 |
. . . . . . . . . . 11
⊢ 𝑒 ∈ V |
55 | 54 | difexi 5235 |
. . . . . . . . . 10
⊢ (𝑒 ∖ 𝐴) ∈ V |
56 | 53, 55 | unex 7472 |
. . . . . . . . 9
⊢ (𝑐 ∪ (𝑒 ∖ 𝐴)) ∈ V |
57 | | sseq1 3995 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (𝑎 ⊆ 𝑋 ↔ (𝑐 ∪ (𝑒 ∖ 𝐴)) ⊆ 𝑋)) |
58 | | sseq2 3996 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (𝑏 ⊆ 𝑎 ↔ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) |
59 | 58 | anbi2d 630 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → ((𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) ↔ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))))) |
60 | 59 | rexbidv 3300 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) ↔ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))))) |
61 | 57, 60 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → ((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ↔ ((𝑐 ∪ (𝑒 ∖ 𝐴)) ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))))) |
62 | | ineq1 4184 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (𝑎 ∩ 𝐴) = ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴)) |
63 | 62 | eqeq2d 2835 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (𝑐 = (𝑎 ∩ 𝐴) ↔ 𝑐 = ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴))) |
64 | 61, 63 | anbi12d 632 |
. . . . . . . . 9
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ↔ (((𝑐 ∪ (𝑒 ∖ 𝐴)) ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) ∧ 𝑐 = ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴)))) |
65 | 56, 64 | spcev 3610 |
. . . . . . . 8
⊢ ((((𝑐 ∪ (𝑒 ∖ 𝐴)) ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) ∧ 𝑐 = ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴)) → ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
66 | 23, 41, 52, 65 | syl21anc 835 |
. . . . . . 7
⊢
(((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ∧ 𝑒 ∈ 𝐽) ∧ 𝑑 = (𝑒 ∩ 𝐴)) → ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
67 | 9 | ad3antrrr 728 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → 𝐽 ∈ Top) |
68 | 9 | uniexd 7471 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ∪ 𝐽 ∈ V) |
69 | 11, 68 | eqeltrid 2920 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝑋 ∈ V) |
70 | 69, 10 | ssexd 5231 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ V) |
71 | 70 | ad3antrrr 728 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → 𝐴 ∈ V) |
72 | | simplr 767 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → 𝑑 ∈ (𝐽 ↾t 𝐴)) |
73 | | elrest 16704 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑑 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑒 ∈ 𝐽 𝑑 = (𝑒 ∩ 𝐴))) |
74 | 73 | biimpa 479 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ V) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) → ∃𝑒 ∈ 𝐽 𝑑 = (𝑒 ∩ 𝐴)) |
75 | 67, 71, 72, 74 | syl21anc 835 |
. . . . . . 7
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → ∃𝑒 ∈ 𝐽 𝑑 = (𝑒 ∩ 𝐴)) |
76 | 66, 75 | r19.29a 3292 |
. . . . . 6
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
77 | 7, 76 | sylanl1 678 |
. . . . 5
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐))) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
78 | | simprr 771 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐))) → ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) |
79 | 5, 77, 78 | r19.29af 3334 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐))) → ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
80 | | inss2 4209 |
. . . . . . . . . 10
⊢ (𝑎 ∩ 𝐴) ⊆ 𝐴 |
81 | | sseq1 3995 |
. . . . . . . . . 10
⊢ (𝑐 = (𝑎 ∩ 𝐴) → (𝑐 ⊆ 𝐴 ↔ (𝑎 ∩ 𝐴) ⊆ 𝐴)) |
82 | 80, 81 | mpbiri 260 |
. . . . . . . . 9
⊢ (𝑐 = (𝑎 ∩ 𝐴) → 𝑐 ⊆ 𝐴) |
83 | 82 | adantl 484 |
. . . . . . . 8
⊢ (((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)) → 𝑐 ⊆ 𝐴) |
84 | 83 | exlimiv 1930 |
. . . . . . 7
⊢
(∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)) → 𝑐 ⊆ 𝐴) |
85 | 84 | adantl 484 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → 𝑐 ⊆ 𝐴) |
86 | 13 | adantr 483 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
87 | 85, 86 | sseqtrd 4010 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) |
88 | 9 | ad4antr 730 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐽 ∈ Top) |
89 | 70 | ad4antr 730 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐴 ∈ V) |
90 | | simplr 767 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝑏 ∈ 𝐽) |
91 | | elrestr 16705 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝑏 ∈ 𝐽) → (𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
92 | 88, 89, 90, 91 | syl3anc 1367 |
. . . . . . . . . . . . . 14
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → (𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
93 | | simprl 769 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐵 ⊆ 𝑏) |
94 | | simp3 1134 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) |
95 | 94 | ad4antr 730 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐵 ⊆ 𝐴) |
96 | 93, 95 | ssind 4212 |
. . . . . . . . . . . . . 14
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐵 ⊆ (𝑏 ∩ 𝐴)) |
97 | | simprr 771 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝑏 ⊆ 𝑎) |
98 | 97 | ssrind 4215 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → (𝑏 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴)) |
99 | | simp-4r 782 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝑐 = (𝑎 ∩ 𝐴)) |
100 | 98, 99 | sseqtrrd 4011 |
. . . . . . . . . . . . . 14
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → (𝑏 ∩ 𝐴) ⊆ 𝑐) |
101 | 92, 96, 100 | jca32 518 |
. . . . . . . . . . . . 13
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐))) |
102 | 101 | ex 415 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) → ((𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) → ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐)))) |
103 | 102 | reximdva 3277 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) → (∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) → ∃𝑏 ∈ 𝐽 ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐)))) |
104 | 103 | impr 457 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ (𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎))) → ∃𝑏 ∈ 𝐽 ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐))) |
105 | 104 | an32s 650 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎))) ∧ 𝑐 = (𝑎 ∩ 𝐴)) → ∃𝑏 ∈ 𝐽 ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐))) |
106 | 105 | expl 460 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)) → ∃𝑏 ∈ 𝐽 ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐)))) |
107 | 106 | exlimdv 1933 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)) → ∃𝑏 ∈ 𝐽 ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐)))) |
108 | 107 | imp 409 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → ∃𝑏 ∈ 𝐽 ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐))) |
109 | | sseq2 3996 |
. . . . . . . . 9
⊢ (𝑑 = (𝑏 ∩ 𝐴) → (𝐵 ⊆ 𝑑 ↔ 𝐵 ⊆ (𝑏 ∩ 𝐴))) |
110 | | sseq1 3995 |
. . . . . . . . 9
⊢ (𝑑 = (𝑏 ∩ 𝐴) → (𝑑 ⊆ 𝑐 ↔ (𝑏 ∩ 𝐴) ⊆ 𝑐)) |
111 | 109, 110 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑑 = (𝑏 ∩ 𝐴) → ((𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐) ↔ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐))) |
112 | 111 | rspcev 3626 |
. . . . . . 7
⊢ (((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) |
113 | 112 | rexlimivw 3285 |
. . . . . 6
⊢
(∃𝑏 ∈
𝐽 ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) |
114 | 108, 113 | syl 17 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) |
115 | 87, 114 | jca 514 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐))) |
116 | 79, 115 | impbida 799 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) ↔ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)))) |
117 | | resttop 21771 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top) |
118 | 9, 70, 117 | syl2anc 586 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝐽 ↾t 𝐴) ∈ Top) |
119 | 94, 13 | sseqtrd 4010 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ ∪ (𝐽 ↾t 𝐴)) |
120 | | eqid 2824 |
. . . . 5
⊢ ∪ (𝐽
↾t 𝐴) =
∪ (𝐽 ↾t 𝐴) |
121 | 120 | isnei 21714 |
. . . 4
⊢ (((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐵 ⊆ ∪ (𝐽
↾t 𝐴))
→ (𝑐 ∈
((nei‘(𝐽
↾t 𝐴))‘𝐵) ↔ (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)))) |
122 | 118, 119,
121 | syl2anc 586 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑐 ∈ ((nei‘(𝐽 ↾t 𝐴))‘𝐵) ↔ (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)))) |
123 | | fvex 6686 |
. . . . . 6
⊢
((nei‘𝐽)‘𝐵) ∈ V |
124 | | restval 16703 |
. . . . . 6
⊢
((((nei‘𝐽)‘𝐵) ∈ V ∧ 𝐴 ∈ V) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴))) |
125 | 123, 70, 124 | sylancr 589 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴))) |
126 | 125 | eleq2d 2901 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ 𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)))) |
127 | 94, 10 | sstrd 3980 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝑋) |
128 | | eqid 2824 |
. . . . . . . . 9
⊢ (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) = (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) |
129 | 128 | elrnmpt 5831 |
. . . . . . . 8
⊢ (𝑐 ∈ V → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎 ∩ 𝐴))) |
130 | 129 | elv 3502 |
. . . . . . 7
⊢ (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎 ∩ 𝐴)) |
131 | | df-rex 3147 |
. . . . . . 7
⊢
(∃𝑎 ∈
((nei‘𝐽)‘𝐵)𝑐 = (𝑎 ∩ 𝐴) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
132 | 130, 131 | bitri 277 |
. . . . . 6
⊢ (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
133 | 11 | isnei 21714 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↔ (𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)))) |
134 | 133 | anbi1d 631 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ↔ ((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)))) |
135 | 134 | exbidv 1921 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → (∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ↔ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)))) |
136 | 132, 135 | syl5bb 285 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) ↔ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)))) |
137 | 9, 127, 136 | syl2anc 586 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) ↔ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)))) |
138 | 126, 137 | bitrd 281 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)))) |
139 | 116, 122,
138 | 3bitr4d 313 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑐 ∈ ((nei‘(𝐽 ↾t 𝐴))‘𝐵) ↔ 𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴))) |
140 | 139 | eqrdv 2822 |
1
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴)) |