Step | Hyp | Ref
| Expression |
1 | | nfv 1843 |
. . . . . 6
⊢
Ⅎ𝑑(𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) |
2 | | nfv 1843 |
. . . . . . 7
⊢
Ⅎ𝑑 𝑐 ⊆ ∪ (𝐽
↾t 𝐴) |
3 | | nfre1 3005 |
. . . . . . 7
⊢
Ⅎ𝑑∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐) |
4 | 2, 3 | nfan 1828 |
. . . . . 6
⊢
Ⅎ𝑑(𝑐 ⊆ ∪ (𝐽
↾t 𝐴)
∧ ∃𝑑 ∈
(𝐽 ↾t
𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) |
5 | 1, 4 | nfan 1828 |
. . . . 5
⊢
Ⅎ𝑑((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑐 ⊆ ∪ (𝐽 ↾t 𝐴) ∧ ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐))) |
6 | | simpl 473 |
. . . . . . 7
⊢ ((𝑐 ⊆ ∪ (𝐽
↾t 𝐴)
∧ ∃𝑑 ∈
(𝐽 ↾t
𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) |
7 | 6 | anim2i 593 |
. . . . . 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 809 |
. . . . . . . . . . 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 1061 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top) |
10 | | simp2 1062 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝑋) |
11 | | neitr.1 |
. . . . . . . . . . . . . 14
⊢ 𝑋 = ∪
𝐽 |
12 | 11 | restuni 20966 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
13 | 9, 10, 12 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
14 | 13 | ad5antr 770 |
. . . . . . . . . . 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 | sseqtr4d 3642 |
. . . . . . . . . 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 770 |
. . . . . . . . . 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 3613 |
. . . . . . . . 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 770 |
. . . . . . . . . . 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 792 |
. . . . . . . . . . 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 20712 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑒 ∈ 𝐽) → 𝑒 ⊆ 𝑋) |
21 | 18, 19, 20 | syl2anc 693 |
. . . . . . . . . 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 3748 |
. . . . . . . . 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 3789 |
. . . . . . . 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 1118 |
. . . . . . . . . . . 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 1290 |
. . . . . . . . . . 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 477 |
. . . . . . . . . . 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 3641 |
. . . . . . . . . 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 3833 |
. . . . . . . . . 10
⊢ (𝑒 ∩ 𝐴) ⊆ 𝑒 |
29 | 27, 28 | syl6ss 3615 |
. . . . . . . . 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 4046 |
. . . . . . . . . 10
⊢ ((𝑒 ∩ 𝐴) ∪ (𝑒 ∖ 𝐴)) = 𝑒 |
31 | | simpr1r 1119 |
. . . . . . . . . . . . 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 1290 |
. . . . . . . . . . . 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 | eqsstr3d 3640 |
. . . . . . . . . . 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 3782 |
. . . . . . . . . . 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 | syl5eqssr 3650 |
. . . . . . . . 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 3627 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑒 → (𝐵 ⊆ 𝑏 ↔ 𝐵 ⊆ 𝑒)) |
38 | | sseq1 3626 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑒 → (𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)) ↔ 𝑒 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) |
39 | 37, 38 | anbi12d 747 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑒 → ((𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))) ↔ (𝐵 ⊆ 𝑒 ∧ 𝑒 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))))) |
40 | 39 | rspcev 3309 |
. . . . . . . . 9
⊢ ((𝑒 ∈ 𝐽 ∧ (𝐵 ⊆ 𝑒 ∧ 𝑒 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) → ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) |
41 | 19, 29, 36, 40 | syl12anc 1324 |
. . . . . . . 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 3875 |
. . . . . . . . . . 11
⊢ ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴) = ((𝑐 ∩ 𝐴) ∪ ((𝑒 ∖ 𝐴) ∩ 𝐴)) |
43 | | incom 3805 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∩ (𝑒 ∖ 𝐴)) = ((𝑒 ∖ 𝐴) ∩ 𝐴) |
44 | | disjdif 4040 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∩ (𝑒 ∖ 𝐴)) = ∅ |
45 | 43, 44 | eqtr3i 2646 |
. . . . . . . . . . . 12
⊢ ((𝑒 ∖ 𝐴) ∩ 𝐴) = ∅ |
46 | 45 | uneq2i 3764 |
. . . . . . . . . . 11
⊢ ((𝑐 ∩ 𝐴) ∪ ((𝑒 ∖ 𝐴) ∩ 𝐴)) = ((𝑐 ∩ 𝐴) ∪ ∅) |
47 | | un0 3967 |
. . . . . . . . . . 11
⊢ ((𝑐 ∩ 𝐴) ∪ ∅) = (𝑐 ∩ 𝐴) |
48 | 42, 46, 47 | 3eqtri 2648 |
. . . . . . . . . 10
⊢ ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴) = (𝑐 ∩ 𝐴) |
49 | | df-ss 3588 |
. . . . . . . . . . 11
⊢ (𝑐 ⊆ 𝐴 ↔ (𝑐 ∩ 𝐴) = 𝑐) |
50 | 49 | biimpi 206 |
. . . . . . . . . 10
⊢ (𝑐 ⊆ 𝐴 → (𝑐 ∩ 𝐴) = 𝑐) |
51 | 48, 50 | syl5req 2669 |
. . . . . . . . 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 3203 |
. . . . . . . . . 10
⊢ 𝑐 ∈ V |
54 | | vex 3203 |
. . . . . . . . . . 11
⊢ 𝑒 ∈ V |
55 | | difexg 4808 |
. . . . . . . . . . 11
⊢ (𝑒 ∈ V → (𝑒 ∖ 𝐴) ∈ V) |
56 | 54, 55 | ax-mp 5 |
. . . . . . . . . 10
⊢ (𝑒 ∖ 𝐴) ∈ V |
57 | 53, 56 | unex 6956 |
. . . . . . . . 9
⊢ (𝑐 ∪ (𝑒 ∖ 𝐴)) ∈ V |
58 | | sseq1 3626 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (𝑎 ⊆ 𝑋 ↔ (𝑐 ∪ (𝑒 ∖ 𝐴)) ⊆ 𝑋)) |
59 | | sseq2 3627 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (𝑏 ⊆ 𝑎 ↔ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴)))) |
60 | 59 | anbi2d 740 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → ((𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) ↔ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))))) |
61 | 60 | rexbidv 3052 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎) ↔ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑐 ∪ (𝑒 ∖ 𝐴))))) |
62 | 58, 61 | anbi12d 747 |
. . . . . . . . . 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)))))) |
63 | | ineq1 3807 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (𝑎 ∩ 𝐴) = ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴)) |
64 | 63 | eqeq2d 2632 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑐 ∪ (𝑒 ∖ 𝐴)) → (𝑐 = (𝑎 ∩ 𝐴) ↔ 𝑐 = ((𝑐 ∪ (𝑒 ∖ 𝐴)) ∩ 𝐴))) |
65 | 62, 64 | anbi12d 747 |
. . . . . . . . 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)))) |
66 | 57, 65 | spcev 3300 |
. . . . . . . 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))) |
67 | 23, 41, 52, 66 | syl21anc 1325 |
. . . . . . 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))) |
68 | 9 | ad3antrrr 766 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → 𝐽 ∈ Top) |
69 | | uniexg 6955 |
. . . . . . . . . . . 12
⊢ (𝐽 ∈ Top → ∪ 𝐽
∈ V) |
70 | 9, 69 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ∪ 𝐽 ∈ V) |
71 | 11, 70 | syl5eqel 2705 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝑋 ∈ V) |
72 | 71, 10 | ssexd 4805 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ V) |
73 | 72 | ad3antrrr 766 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → 𝐴 ∈ V) |
74 | | simplr 792 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → 𝑑 ∈ (𝐽 ↾t 𝐴)) |
75 | | elrest 16088 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑑 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑒 ∈ 𝐽 𝑑 = (𝑒 ∩ 𝐴))) |
76 | 75 | biimpa 501 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ V) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) → ∃𝑒 ∈ 𝐽 𝑑 = (𝑒 ∩ 𝐴)) |
77 | 68, 73, 74, 76 | syl21anc 1325 |
. . . . . . 7
⊢
(((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) ∧ 𝑑 ∈ (𝐽 ↾t 𝐴)) ∧ (𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) → ∃𝑒 ∈ 𝐽 𝑑 = (𝑒 ∩ 𝐴)) |
78 | 67, 77 | r19.29a 3078 |
. . . . . 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))) |
79 | 7, 78 | sylanl1 682 |
. . . . 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))) |
80 | | simprr 796 |
. . . . 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)) |
81 | 5, 79, 80 | r19.29af 3076 |
. . . 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))) |
82 | | inss2 3834 |
. . . . . . . . . 10
⊢ (𝑎 ∩ 𝐴) ⊆ 𝐴 |
83 | | sseq1 3626 |
. . . . . . . . . 10
⊢ (𝑐 = (𝑎 ∩ 𝐴) → (𝑐 ⊆ 𝐴 ↔ (𝑎 ∩ 𝐴) ⊆ 𝐴)) |
84 | 82, 83 | mpbiri 248 |
. . . . . . . . 9
⊢ (𝑐 = (𝑎 ∩ 𝐴) → 𝑐 ⊆ 𝐴) |
85 | 84 | adantl 482 |
. . . . . . . 8
⊢ (((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)) → 𝑐 ⊆ 𝐴) |
86 | 85 | exlimiv 1858 |
. . . . . . 7
⊢
(∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)) → 𝑐 ⊆ 𝐴) |
87 | 86 | adantl 482 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → 𝑐 ⊆ 𝐴) |
88 | 13 | adantr 481 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
89 | 87, 88 | sseqtrd 3641 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑎((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴))) → 𝑐 ⊆ ∪ (𝐽 ↾t 𝐴)) |
90 | 9 | ad4antr 768 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐽 ∈ Top) |
91 | 72 | ad4antr 768 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐴 ∈ V) |
92 | | simplr 792 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝑏 ∈ 𝐽) |
93 | | elrestr 16089 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝑏 ∈ 𝐽) → (𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
94 | 90, 91, 92, 93 | syl3anc 1326 |
. . . . . . . . . . . . . 14
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → (𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
95 | | simprl 794 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐵 ⊆ 𝑏) |
96 | | simp3 1063 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) |
97 | 96 | ad4antr 768 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐵 ⊆ 𝐴) |
98 | 95, 97 | ssind 3837 |
. . . . . . . . . . . . . 14
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝐵 ⊆ (𝑏 ∩ 𝐴)) |
99 | | simprr 796 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝑏 ⊆ 𝑎) |
100 | | ssrin 3838 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ⊆ 𝑎 → (𝑏 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴)) |
101 | 99, 100 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → (𝑏 ∩ 𝐴) ⊆ (𝑎 ∩ 𝐴)) |
102 | | simp-4r 807 |
. . . . . . . . . . . . . . 15
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → 𝑐 = (𝑎 ∩ 𝐴)) |
103 | 101, 102 | sseqtr4d 3642 |
. . . . . . . . . . . . . 14
⊢
((((((𝐽 ∈ Top
∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ∧ 𝑎 ⊆ 𝑋) ∧ 𝑏 ∈ 𝐽) ∧ (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) → (𝑏 ∩ 𝐴) ⊆ 𝑐) |
104 | 94, 98, 103 | jca32 558 |
. . . . . . . . . . . . 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))) |
105 | 104 | ex 450 |
. . . . . . . . . . . 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)))) |
106 | 105 | reximdva 3017 |
. . . . . . . . . . 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)))) |
107 | 106 | impr 649 |
. . . . . . . . . 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))) |
108 | 107 | an32s 846 |
. . . . . . . . 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))) |
109 | 108 | expl 648 |
. . . . . . . 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)))) |
110 | 109 | exlimdv 1861 |
. . . . . . 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)))) |
111 | 110 | imp 445 |
. . . . . 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))) |
112 | | sseq2 3627 |
. . . . . . . . 9
⊢ (𝑑 = (𝑏 ∩ 𝐴) → (𝐵 ⊆ 𝑑 ↔ 𝐵 ⊆ (𝑏 ∩ 𝐴))) |
113 | | sseq1 3626 |
. . . . . . . . 9
⊢ (𝑑 = (𝑏 ∩ 𝐴) → (𝑑 ⊆ 𝑐 ↔ (𝑏 ∩ 𝐴) ⊆ 𝑐)) |
114 | 112, 113 | anbi12d 747 |
. . . . . . . 8
⊢ (𝑑 = (𝑏 ∩ 𝐴) → ((𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐) ↔ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐))) |
115 | 114 | rspcev 3309 |
. . . . . . 7
⊢ (((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) |
116 | 115 | rexlimivw 3029 |
. . . . . 6
⊢
(∃𝑏 ∈
𝐽 ((𝑏 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴) ∧ (𝐵 ⊆ (𝑏 ∩ 𝐴) ∧ (𝑏 ∩ 𝐴) ⊆ 𝑐)) → ∃𝑑 ∈ (𝐽 ↾t 𝐴)(𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐)) |
117 | 111, 116 | 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)) |
118 | 89, 117 | jca 554 |
. . . 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))) |
119 | 81, 118 | impbida 877 |
. . 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)))) |
120 | | resttop 20964 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top) |
121 | 9, 72, 120 | syl2anc 693 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝐽 ↾t 𝐴) ∈ Top) |
122 | 96, 13 | sseqtrd 3641 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ ∪ (𝐽 ↾t 𝐴)) |
123 | | eqid 2622 |
. . . . 5
⊢ ∪ (𝐽
↾t 𝐴) =
∪ (𝐽 ↾t 𝐴) |
124 | 123 | isnei 20907 |
. . . 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)))) |
125 | 121, 122,
124 | syl2anc 693 |
. . 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)))) |
126 | | fvex 6201 |
. . . . . 6
⊢
((nei‘𝐽)‘𝐵) ∈ V |
127 | | restval 16087 |
. . . . . 6
⊢
((((nei‘𝐽)‘𝐵) ∈ V ∧ 𝐴 ∈ V) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴))) |
128 | 126, 72, 127 | sylancr 695 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (((nei‘𝐽)‘𝐵) ↾t 𝐴) = ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴))) |
129 | 128 | eleq2d 2687 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴) ↔ 𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)))) |
130 | 96, 10 | sstrd 3613 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝑋) |
131 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) = (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) |
132 | 131 | elrnmpt 5372 |
. . . . . . . 8
⊢ (𝑐 ∈ V → (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎 ∩ 𝐴))) |
133 | 53, 132 | ax-mp 5 |
. . . . . . 7
⊢ (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) ↔ ∃𝑎 ∈ ((nei‘𝐽)‘𝐵)𝑐 = (𝑎 ∩ 𝐴)) |
134 | | df-rex 2918 |
. . . . . . 7
⊢
(∃𝑎 ∈
((nei‘𝐽)‘𝐵)𝑐 = (𝑎 ∩ 𝐴) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
135 | 133, 134 | bitri 264 |
. . . . . 6
⊢ (𝑐 ∈ ran (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↦ (𝑎 ∩ 𝐴)) ↔ ∃𝑎(𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎 ∩ 𝐴))) |
136 | 11 | isnei 20907 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → (𝑎 ∈ ((nei‘𝐽)‘𝐵) ↔ (𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)))) |
137 | 136 | anbi1d 741 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((𝑎 ∈ ((nei‘𝐽)‘𝐵) ∧ 𝑐 = (𝑎 ∩ 𝐴)) ↔ ((𝑎 ⊆ 𝑋 ∧ ∃𝑏 ∈ 𝐽 (𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎)) ∧ 𝑐 = (𝑎 ∩ 𝐴)))) |
138 | 137 | exbidv 1850 |
. . . . . 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)))) |
139 | 135, 138 | syl5bb 272 |
. . . . 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)))) |
140 | 9, 130, 139 | syl2anc 693 |
. . . 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)))) |
141 | 129, 140 | bitrd 268 |
. . 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)))) |
142 | 119, 125,
141 | 3bitr4d 300 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → (𝑐 ∈ ((nei‘(𝐽 ↾t 𝐴))‘𝐵) ↔ 𝑐 ∈ (((nei‘𝐽)‘𝐵) ↾t 𝐴))) |
143 | 142 | eqrdv 2620 |
1
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘𝐵) = (((nei‘𝐽)‘𝐵) ↾t 𝐴)) |