Step | Hyp | Ref
| Expression |
1 | | methaus.1 |
. . 3
⊢ 𝐽 = (MetOpen‘𝐷) |
2 | 1 | mopntop 23574 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
3 | 1 | mopnuni 23575 |
. . . . . 6
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
4 | 3 | eleq2d 2825 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽)) |
5 | 4 | biimpar 477 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋) |
6 | | simpll 763 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) |
7 | | simplr 765 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝑋) |
8 | | nnrp 12723 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
9 | 8 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+) |
10 | 9 | rpreccld 12764 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ+) |
11 | 10 | rpxrd 12755 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ*) |
12 | 1 | blopn 23637 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ*) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽) |
13 | 6, 7, 11, 12 | syl3anc 1369 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽) |
14 | 13 | fmpttd 6983 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ⟶𝐽) |
15 | 14 | frnd 6604 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽) |
16 | | nnex 11962 |
. . . . . . . . 9
⊢ ℕ
∈ V |
17 | 16 | mptex 7093 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V |
18 | 17 | rnex 7746 |
. . . . . . 7
⊢ ran
(𝑛 ∈ ℕ ↦
(𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V |
19 | 18 | elpw 4542 |
. . . . . 6
⊢ (ran
(𝑛 ∈ ℕ ↦
(𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽 ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽) |
20 | 15, 19 | sylibr 233 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽) |
21 | | omelon 9365 |
. . . . . . . . 9
⊢ ω
∈ On |
22 | | nnenom 13681 |
. . . . . . . . . 10
⊢ ℕ
≈ ω |
23 | 22 | ensymi 8761 |
. . . . . . . . 9
⊢ ω
≈ ℕ |
24 | | isnumi 9688 |
. . . . . . . . 9
⊢ ((ω
∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom
card) |
25 | 21, 23, 24 | mp2an 688 |
. . . . . . . 8
⊢ ℕ
∈ dom card |
26 | | ovex 7301 |
. . . . . . . . . 10
⊢ (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ V |
27 | | eqid 2739 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) |
28 | 26, 27 | fnmpti 6572 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) Fn ℕ |
29 | | dffn4 6690 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) Fn ℕ ↔ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ–onto→ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))) |
30 | 28, 29 | mpbi 229 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ–onto→ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) |
31 | | fodomnum 9797 |
. . . . . . . 8
⊢ (ℕ
∈ dom card → ((𝑛
∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ–onto→ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ℕ)) |
32 | 25, 30, 31 | mp2 9 |
. . . . . . 7
⊢ ran
(𝑛 ∈ ℕ ↦
(𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ℕ |
33 | | domentr 8770 |
. . . . . . 7
⊢ ((ran
(𝑛 ∈ ℕ ↦
(𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ℕ ∧ ℕ ≈
ω) → ran (𝑛
∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω) |
34 | 32, 22, 33 | mp2an 688 |
. . . . . 6
⊢ ran
(𝑛 ∈ ℕ ↦
(𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω |
35 | 34 | a1i 11 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω) |
36 | | simpll 763 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝐷 ∈ (∞Met‘𝑋)) |
37 | | simprl 767 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ∈ 𝐽) |
38 | | simprr 769 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ 𝑧) |
39 | 1 | mopni2 23630 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧) |
40 | 36, 37, 38, 39 | syl3anc 1369 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧) |
41 | | simp-4l 779 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝐷 ∈ (∞Met‘𝑋)) |
42 | | simp-4r 780 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑥 ∈ 𝑋) |
43 | | simprl 767 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑦 ∈ ℕ) |
44 | 43 | nnrpd 12752 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑦 ∈ ℝ+) |
45 | 44 | rpreccld 12764 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) ∈
ℝ+) |
46 | | blcntr 23547 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑦) ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦))) |
47 | 41, 42, 45, 46 | syl3anc 1369 |
. . . . . . . . . . 11
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦))) |
48 | 45 | rpxrd 12755 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) ∈
ℝ*) |
49 | | simplrl 773 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑟 ∈ ℝ+) |
50 | 49 | rpxrd 12755 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑟 ∈ ℝ*) |
51 | | nnrecre 11998 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℕ → (1 /
𝑦) ∈
ℝ) |
52 | 51 | ad2antrl 724 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) ∈ ℝ) |
53 | 49 | rpred 12754 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → 𝑟 ∈ ℝ) |
54 | | simprr 769 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) < 𝑟) |
55 | 52, 53, 54 | ltled 11106 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (1 / 𝑦) ≤ 𝑟) |
56 | | ssbl 23557 |
. . . . . . . . . . . . 13
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ ((1 / 𝑦) ∈ ℝ* ∧ 𝑟 ∈ ℝ*)
∧ (1 / 𝑦) ≤ 𝑟) → (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ (𝑥(ball‘𝐷)𝑟)) |
57 | 41, 42, 48, 50, 55, 56 | syl221anc 1379 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ (𝑥(ball‘𝐷)𝑟)) |
58 | | simplrr 774 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧) |
59 | 57, 58 | sstrd 3935 |
. . . . . . . . . . 11
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧) |
60 | 47, 59 | jca 511 |
. . . . . . . . . 10
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < 𝑟)) → (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)) |
61 | | elrp 12714 |
. . . . . . . . . . . 12
⊢ (𝑟 ∈ ℝ+
↔ (𝑟 ∈ ℝ
∧ 0 < 𝑟)) |
62 | | nnrecl 12214 |
. . . . . . . . . . . 12
⊢ ((𝑟 ∈ ℝ ∧ 0 <
𝑟) → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟) |
63 | 61, 62 | sylbi 216 |
. . . . . . . . . . 11
⊢ (𝑟 ∈ ℝ+
→ ∃𝑦 ∈
ℕ (1 / 𝑦) < 𝑟) |
64 | 63 | ad2antrl 724 |
. . . . . . . . . 10
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟) |
65 | 60, 64 | reximddv 3205 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧)) → ∃𝑦 ∈ ℕ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)) |
66 | 40, 65 | rexlimddv 3221 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → ∃𝑦 ∈ ℕ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)) |
67 | | ovexd 7303 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ 𝑦 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑦)) ∈ V) |
68 | | vex 3434 |
. . . . . . . . . 10
⊢ 𝑤 ∈ V |
69 | | oveq2 7276 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑦 → (1 / 𝑛) = (1 / 𝑦)) |
70 | 69 | oveq2d 7284 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑦 → (𝑥(ball‘𝐷)(1 / 𝑛)) = (𝑥(ball‘𝐷)(1 / 𝑦))) |
71 | 70 | cbvmptv 5191 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑦 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑦))) |
72 | 71 | elrnmpt 5862 |
. . . . . . . . . 10
⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ↔ ∃𝑦 ∈ ℕ 𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)))) |
73 | 68, 72 | mp1i 13 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → (𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ↔ ∃𝑦 ∈ ℕ 𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)))) |
74 | | eleq2 2828 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)))) |
75 | | sseq1 3950 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑤 ⊆ 𝑧 ↔ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)) |
76 | 74, 75 | anbi12d 630 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))) |
77 | 76 | adantl 481 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ 𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦))) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))) |
78 | 67, 73, 77 | rexxfr2d 5337 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → (∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑦 ∈ ℕ (𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)) ∧ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧))) |
79 | 66, 78 | mpbird 256 |
. . . . . . 7
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) |
80 | 79 | expr 456 |
. . . . . 6
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝐽) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) |
81 | 80 | ralrimiva 3109 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) |
82 | | breq1 5081 |
. . . . . . 7
⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (𝑦 ≼ ω ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω)) |
83 | | rexeq 3341 |
. . . . . . . . 9
⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) |
84 | 83 | imbi2d 340 |
. . . . . . . 8
⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → ((𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) |
85 | 84 | ralbidv 3122 |
. . . . . . 7
⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) |
86 | 82, 85 | anbi12d 630 |
. . . . . 6
⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ↔ (ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) |
87 | 86 | rspcev 3560 |
. . . . 5
⊢ ((ran
(𝑛 ∈ ℕ ↦
(𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽 ∧ (ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛)))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) |
88 | 20, 35, 81, 87 | syl12anc 833 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) |
89 | 5, 88 | syldan 590 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) |
90 | 89 | ralrimiva 3109 |
. 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) |
91 | | eqid 2739 |
. . 3
⊢ ∪ 𝐽 =
∪ 𝐽 |
92 | 91 | is1stc2 22574 |
. 2
⊢ (𝐽 ∈ 1stω
↔ (𝐽 ∈ Top ∧
∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) |
93 | 2, 90, 92 | sylanbrc 582 |
1
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈
1stω) |