| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | methaus.1 | . . 3
⊢ 𝐽 = (MetOpen‘𝐷) | 
| 2 | 1 | mopntop 24450 | . 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) | 
| 3 | 1 | mopnuni 24451 | . . . . . 6
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 4 | 3 | eleq2d 2827 | . . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽)) | 
| 5 | 4 | biimpar 477 | . . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → 𝑥 ∈ 𝑋) | 
| 6 |  | simpll 767 | . . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) | 
| 7 |  | simplr 769 | . . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ 𝑋) | 
| 8 |  | nnrp 13046 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) | 
| 9 | 8 | adantl 481 | . . . . . . . . . . 11
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+) | 
| 10 | 9 | rpreccld 13087 | . . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ+) | 
| 11 | 10 | rpxrd 13078 | . . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈
ℝ*) | 
| 12 | 1 | blopn 24513 | . . . . . . . . 9
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑛) ∈ ℝ*) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽) | 
| 13 | 6, 7, 11, 12 | syl3anc 1373 | . . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑛 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ 𝐽) | 
| 14 | 13 | fmpttd 7135 | . . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ⟶𝐽) | 
| 15 | 14 | frnd 6744 | . . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽) | 
| 16 |  | nnex 12272 | . . . . . . . . 9
⊢ ℕ
∈ V | 
| 17 | 16 | mptex 7243 | . . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V | 
| 18 | 17 | rnex 7932 | . . . . . . 7
⊢ ran
(𝑛 ∈ ℕ ↦
(𝑥(ball‘𝐷)(1 / 𝑛))) ∈ V | 
| 19 | 18 | elpw 4604 | . . . . . 6
⊢ (ran
(𝑛 ∈ ℕ ↦
(𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽 ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ⊆ 𝐽) | 
| 20 | 15, 19 | sylibr 234 | . . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ∈ 𝒫 𝐽) | 
| 21 |  | omelon 9686 | . . . . . . . . 9
⊢ ω
∈ On | 
| 22 |  | nnenom 14021 | . . . . . . . . . 10
⊢ ℕ
≈ ω | 
| 23 | 22 | ensymi 9044 | . . . . . . . . 9
⊢ ω
≈ ℕ | 
| 24 |  | isnumi 9986 | . . . . . . . . 9
⊢ ((ω
∈ On ∧ ω ≈ ℕ) → ℕ ∈ dom
card) | 
| 25 | 21, 23, 24 | mp2an 692 | . . . . . . . 8
⊢ ℕ
∈ dom card | 
| 26 |  | ovex 7464 | . . . . . . . . . 10
⊢ (𝑥(ball‘𝐷)(1 / 𝑛)) ∈ V | 
| 27 |  | eqid 2737 | . . . . . . . . . 10
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) | 
| 28 | 26, 27 | fnmpti 6711 | . . . . . . . . 9
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) Fn ℕ | 
| 29 |  | dffn4 6826 | . . . . . . . . 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 230 | . . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))):ℕ–onto→ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) | 
| 31 |  | fodomnum 10097 | . . . . . . . 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 9053 | . . . . . . 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 692 | . . . . . 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 767 | . . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝐷 ∈ (∞Met‘𝑋)) | 
| 37 |  | simprl 771 | . . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ∈ 𝐽) | 
| 38 |  | simprr 773 | . . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ 𝑧) | 
| 39 | 1 | mopni2 24506 | . . . . . . . . . 10
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧) | 
| 40 | 36, 37, 38, 39 | syl3anc 1373 | . . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐷)𝑟) ⊆ 𝑧) | 
| 41 |  | simp-4l 783 | . . . . . . . . . . . 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 784 | . . . . . . . . . . . 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 771 | . . . . . . . . . . . . . 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 13075 | . . . . . . . . . . . . 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 13087 | . . . . . . . . . . . 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 24423 | . . . . . . . . . . . 12
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (1 / 𝑦) ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦))) | 
| 47 | 41, 42, 45, 46 | syl3anc 1373 | . . . . . . . . . . 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 13078 | . . . . . . . . . . . . 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 777 | . . . . . . . . . . . . . 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 13078 | . . . . . . . . . . . . 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 12308 | . . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℕ → (1 /
𝑦) ∈
ℝ) | 
| 52 | 51 | ad2antrl 728 | . . . . . . . . . . . . . 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 13077 | . . . . . . . . . . . . . 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 773 | . . . . . . . . . . . . . 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 11409 | . . . . . . . . . . . . 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 24433 | . . . . . . . . . . . . 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 1383 | . . . . . . . . . . . 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 778 | . . . . . . . . . . . 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 3994 | . . . . . . . . . . 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 13036 | . . . . . . . . . . . 12
⊢ (𝑟 ∈ ℝ+
↔ (𝑟 ∈ ℝ
∧ 0 < 𝑟)) | 
| 62 |  | nnrecl 12524 | . . . . . . . . . . . 12
⊢ ((𝑟 ∈ ℝ ∧ 0 <
𝑟) → ∃𝑦 ∈ ℕ (1 / 𝑦) < 𝑟) | 
| 63 | 61, 62 | sylbi 217 | . . . . . . . . . . 11
⊢ (𝑟 ∈ ℝ+
→ ∃𝑦 ∈
ℕ (1 / 𝑦) < 𝑟) | 
| 64 | 63 | ad2antrl 728 | . . . . . . . . . 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 3171 | . . . . . . . . 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 3161 | . . . . . . . 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 7466 | . . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 ∈ 𝐽 ∧ 𝑥 ∈ 𝑧)) ∧ 𝑦 ∈ ℕ) → (𝑥(ball‘𝐷)(1 / 𝑦)) ∈ V) | 
| 68 |  | vex 3484 | . . . . . . . . . 10
⊢ 𝑤 ∈ V | 
| 69 |  | oveq2 7439 | . . . . . . . . . . . . 13
⊢ (𝑛 = 𝑦 → (1 / 𝑛) = (1 / 𝑦)) | 
| 70 | 69 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ (𝑛 = 𝑦 → (𝑥(ball‘𝐷)(1 / 𝑛)) = (𝑥(ball‘𝐷)(1 / 𝑦))) | 
| 71 | 70 | cbvmptv 5255 | . . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) = (𝑦 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑦))) | 
| 72 | 71 | elrnmpt 5969 | . . . . . . . . . 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 2830 | . . . . . . . . . . 11
⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑥(ball‘𝐷)(1 / 𝑦)))) | 
| 75 |  | sseq1 4009 | . . . . . . . . . . 11
⊢ (𝑤 = (𝑥(ball‘𝐷)(1 / 𝑦)) → (𝑤 ⊆ 𝑧 ↔ (𝑥(ball‘𝐷)(1 / 𝑦)) ⊆ 𝑧)) | 
| 76 | 74, 75 | anbi12d 632 | . . . . . . . . . 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 5411 | . . . . . . . 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 257 | . . . . . . 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 3146 | . . . . 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 5146 | . . . . . . 7
⊢ (𝑦 = ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) → (𝑦 ≼ ω ↔ ran (𝑛 ∈ ℕ ↦ (𝑥(ball‘𝐷)(1 / 𝑛))) ≼ ω)) | 
| 83 |  | rexeq 3322 | . . . . . . . . 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 3178 | . . . . . . 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 632 | . . . . . 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 3622 | . . . . 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 837 | . . . 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 591 | . . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ ∪ 𝐽) → ∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) | 
| 90 | 89 | ralrimiva 3146 | . 2
⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) | 
| 91 |  | eqid 2737 | . . 3
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 92 | 91 | is1stc2 23450 | . 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 583 | 1
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈
1stω) |