Step | Hyp | Ref
| Expression |
1 | | istotbnd 33900 |
. 2
⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑤 ∈ Fin (∪ 𝑤 =
𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
2 | | oveq1 6800 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑓‘𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
3 | 2 | eqeq2d 2781 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑓‘𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
4 | 3 | ac6sfi 8360 |
. . . . . . . . . 10
⊢ ((𝑤 ∈ Fin ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
5 | 4 | ex 397 |
. . . . . . . . 9
⊢ (𝑤 ∈ Fin →
(∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑓(𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) |
6 | 5 | ad2antlr 706 |
. . . . . . . 8
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ∪ 𝑤 =
𝑋) → (∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑓(𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) |
7 | | simprrl 766 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤⟶𝑋) |
8 | | frn 6193 |
. . . . . . . . . . . . 13
⊢ (𝑓:𝑤⟶𝑋 → ran 𝑓 ⊆ 𝑋) |
9 | 7, 8 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ⊆ 𝑋) |
10 | | simplr 752 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑤 ∈ Fin) |
11 | | ffn 6185 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:𝑤⟶𝑋 → 𝑓 Fn 𝑤) |
12 | 7, 11 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓 Fn 𝑤) |
13 | | dffn4 6262 |
. . . . . . . . . . . . . 14
⊢ (𝑓 Fn 𝑤 ↔ 𝑓:𝑤–onto→ran 𝑓) |
14 | 12, 13 | sylib 208 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤–onto→ran 𝑓) |
15 | | fofi 8408 |
. . . . . . . . . . . . 13
⊢ ((𝑤 ∈ Fin ∧ 𝑓:𝑤–onto→ran 𝑓) → ran 𝑓 ∈ Fin) |
16 | 10, 14, 15 | syl2anc 573 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ Fin) |
17 | | elfpw 8424 |
. . . . . . . . . . . 12
⊢ (ran
𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑋 ∧ ran 𝑓 ∈ Fin)) |
18 | 9, 16, 17 | sylanbrc 572 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin)) |
19 | 2 | eleq2d 2836 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑓‘𝑏) → (𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
20 | 19 | rexrn 6504 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 Fn 𝑤 → (∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑤 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
21 | 12, 20 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → (∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑤 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
22 | | eliun 4658 |
. . . . . . . . . . . . . 14
⊢ (𝑣 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑)) |
23 | | eliun 4658 |
. . . . . . . . . . . . . 14
⊢ (𝑣 ∈ ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑤 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
24 | 21, 22, 23 | 3bitr4g 303 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → (𝑣 ∈ ∪
𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑣 ∈ ∪
𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
25 | 24 | eqrdv 2769 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
26 | | simprrr 767 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
27 | | iuneq2 4671 |
. . . . . . . . . . . . 13
⊢
(∀𝑏 ∈
𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪
𝑏 ∈ 𝑤 𝑏 = ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
28 | 26, 27 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑏 ∈ 𝑤 𝑏 = ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
29 | | uniiun 4707 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑤 =
∪ 𝑏 ∈ 𝑤 𝑏 |
30 | | simprl 754 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪
𝑤 = 𝑋) |
31 | 29, 30 | syl5eqr 2819 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑏 ∈ 𝑤 𝑏 = 𝑋) |
32 | 25, 28, 31 | 3eqtr2d 2811 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋) |
33 | | iuneq1 4668 |
. . . . . . . . . . . . 13
⊢ (𝑣 = ran 𝑓 → ∪
𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)) |
34 | 33 | eqeq1d 2773 |
. . . . . . . . . . . 12
⊢ (𝑣 = ran 𝑓 → (∪
𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∪
𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋)) |
35 | 34 | rspcev 3460 |
. . . . . . . . . . 11
⊢ ((ran
𝑓 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋) |
36 | 18, 32, 35 | syl2anc 573 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋) |
37 | 36 | expr 444 |
. . . . . . . . 9
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ∪ 𝑤 =
𝑋) → ((𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |
38 | 37 | exlimdv 2013 |
. . . . . . . 8
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ∪ 𝑤 =
𝑋) → (∃𝑓(𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |
39 | 6, 38 | syld 47 |
. . . . . . 7
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ∪ 𝑤 =
𝑋) → (∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |
40 | 39 | expimpd 441 |
. . . . . 6
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) → ((∪ 𝑤 =
𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |
41 | 40 | rexlimdva 3179 |
. . . . 5
⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑤 ∈ Fin (∪ 𝑤 =
𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |
42 | | elfpw 8424 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ⊆ 𝑋 ∧ 𝑣 ∈ Fin)) |
43 | 42 | simprbi 484 |
. . . . . . . . . 10
⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin) |
44 | 43 | ad2antrl 707 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ∈ Fin) |
45 | | mptfi 8421 |
. . . . . . . . 9
⊢ (𝑣 ∈ Fin → (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin) |
46 | | rnfi 8405 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin → ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin) |
47 | 44, 45, 46 | 3syl 18 |
. . . . . . . 8
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin) |
48 | | ovex 6823 |
. . . . . . . . . 10
⊢ (𝑥(ball‘𝑀)𝑑) ∈ V |
49 | 48 | dfiun3 5518 |
. . . . . . . . 9
⊢ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) |
50 | | simprr 756 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋) |
51 | 49, 50 | syl5eqr 2819 |
. . . . . . . 8
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∪ ran
(𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋) |
52 | | eqid 2771 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) |
53 | 52 | rnmpt 5509 |
. . . . . . . . 9
⊢ ran
(𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)} |
54 | 42 | simplbi 485 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ⊆ 𝑋) |
55 | 54 | ad2antrl 707 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ⊆ 𝑋) |
56 | | ssrexv 3816 |
. . . . . . . . . . 11
⊢ (𝑣 ⊆ 𝑋 → (∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) |
57 | 55, 56 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → (∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) |
58 | 57 | ss2abdv 3824 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → {𝑏 ∣ ∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)} ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}) |
59 | 53, 58 | syl5eqss 3798 |
. . . . . . . 8
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}) |
60 | | unieq 4582 |
. . . . . . . . . . 11
⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑤 = ∪
ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑))) |
61 | 60 | eqeq1d 2773 |
. . . . . . . . . 10
⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (∪ 𝑤 = 𝑋 ↔ ∪ ran
(𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋)) |
62 | | ssabral 3822 |
. . . . . . . . . . 11
⊢ (𝑤 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) |
63 | | sseq1 3775 |
. . . . . . . . . . 11
⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑤 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})) |
64 | 62, 63 | syl5bbr 274 |
. . . . . . . . . 10
⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})) |
65 | 61, 64 | anbi12d 616 |
. . . . . . . . 9
⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → ((∪
𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ (∪ ran
(𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋 ∧ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))) |
66 | 65 | rspcev 3460 |
. . . . . . . 8
⊢ ((ran
(𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin ∧ (∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋 ∧ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})) → ∃𝑤 ∈ Fin (∪
𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) |
67 | 47, 51, 59, 66 | syl12anc 1474 |
. . . . . . 7
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∃𝑤 ∈ Fin (∪
𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) |
68 | 67 | expr 444 |
. . . . . 6
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑤 ∈ Fin (∪
𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
69 | 68 | rexlimdva 3179 |
. . . . 5
⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑤 ∈ Fin (∪
𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
70 | 41, 69 | impbid 202 |
. . . 4
⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑤 ∈ Fin (∪ 𝑤 =
𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |
71 | 70 | ralbidv 3135 |
. . 3
⊢ (𝑀 ∈ (Met‘𝑋) → (∀𝑑 ∈ ℝ+
∃𝑤 ∈ Fin (∪ 𝑤 =
𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |
72 | 71 | pm5.32i 564 |
. 2
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+
∃𝑤 ∈ Fin (∪ 𝑤 =
𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |
73 | 1, 72 | bitri 264 |
1
⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |