| Step | Hyp | Ref
| Expression |
| 1 | | istotbnd 37776 |
. 2
⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑤 ∈ Fin (∪ 𝑤 =
𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) |
| 2 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑓‘𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
| 3 | 2 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑓‘𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
| 4 | 3 | ac6sfi 9320 |
. . . . . . . . . 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 412 |
. . . . . . . . 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 727 |
. . . . . . . 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 781 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤⟶𝑋) |
| 8 | 7 | frnd 6744 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ⊆ 𝑋) |
| 9 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑤 ∈ Fin) |
| 10 | 7 | ffnd 6737 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓 Fn 𝑤) |
| 11 | | dffn4 6826 |
. . . . . . . . . . . . . 14
⊢ (𝑓 Fn 𝑤 ↔ 𝑓:𝑤–onto→ran 𝑓) |
| 12 | 10, 11 | sylib 218 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤–onto→ran 𝑓) |
| 13 | | fofi 9351 |
. . . . . . . . . . . . 13
⊢ ((𝑤 ∈ Fin ∧ 𝑓:𝑤–onto→ran 𝑓) → ran 𝑓 ∈ Fin) |
| 14 | 9, 12, 13 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ Fin) |
| 15 | | elfpw 9394 |
. . . . . . . . . . . 12
⊢ (ran
𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑋 ∧ ran 𝑓 ∈ Fin)) |
| 16 | 8, 14, 15 | sylanbrc 583 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin)) |
| 17 | 2 | eleq2d 2827 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑓‘𝑏) → (𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
| 18 | 17 | rexrn 7107 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 Fn 𝑤 → (∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑤 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))) |
| 19 | 10, 18 | 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))) |
| 20 | | eliun 4995 |
. . . . . . . . . . . . . 14
⊢ (𝑣 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑)) |
| 21 | | eliun 4995 |
. . . . . . . . . . . . . 14
⊢ (𝑣 ∈ ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑤 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
| 22 | 19, 20, 21 | 3bitr4g 314 |
. . . . . . . . . . . . 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))) |
| 23 | 22 | eqrdv 2735 |
. . . . . . . . . . . 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)) |
| 24 | | simprrr 782 |
. . . . . . . . . . . . 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)) |
| 25 | | iuneq2 5011 |
. . . . . . . . . . . . 13
⊢
(∀𝑏 ∈
𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪
𝑏 ∈ 𝑤 𝑏 = ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑)) |
| 26 | 24, 25 | 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)) |
| 27 | | uniiun 5058 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑤 =
∪ 𝑏 ∈ 𝑤 𝑏 |
| 28 | | simprl 771 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪
𝑤 = 𝑋) |
| 29 | 27, 28 | eqtr3id 2791 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 =
𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑏 ∈ 𝑤 𝑏 = 𝑋) |
| 30 | 23, 26, 29 | 3eqtr2d 2783 |
. . . . . . . . . . 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) |
| 31 | | iuneq1 5008 |
. . . . . . . . . . . . 13
⊢ (𝑣 = ran 𝑓 → ∪
𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)) |
| 32 | 31 | eqeq1d 2739 |
. . . . . . . . . . . 12
⊢ (𝑣 = ran 𝑓 → (∪
𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∪
𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋)) |
| 33 | 32 | rspcev 3622 |
. . . . . . . . . . 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) |
| 34 | 16, 30, 33 | syl2anc 584 |
. . . . . . . . . 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) |
| 35 | 34 | expr 456 |
. . . . . . . . 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)) |
| 36 | 35 | exlimdv 1933 |
. . . . . . . 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)) |
| 37 | 6, 36 | 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)) |
| 38 | 37 | expimpd 453 |
. . . . . 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)) |
| 39 | 38 | rexlimdva 3155 |
. . . . 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)) |
| 40 | | elfpw 9394 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ⊆ 𝑋 ∧ 𝑣 ∈ Fin)) |
| 41 | 40 | simprbi 496 |
. . . . . . . . . 10
⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin) |
| 42 | 41 | ad2antrl 728 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ∈ Fin) |
| 43 | | mptfi 9391 |
. . . . . . . . 9
⊢ (𝑣 ∈ Fin → (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin) |
| 44 | | rnfi 9380 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin → ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin) |
| 45 | 42, 43, 44 | 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) |
| 46 | | ovex 7464 |
. . . . . . . . . 10
⊢ (𝑥(ball‘𝑀)𝑑) ∈ V |
| 47 | 46 | dfiun3 5980 |
. . . . . . . . 9
⊢ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) |
| 48 | | simprr 773 |
. . . . . . . . 9
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋) |
| 49 | 47, 48 | eqtr3id 2791 |
. . . . . . . 8
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∪ ran
(𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋) |
| 50 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) |
| 51 | 50 | rnmpt 5968 |
. . . . . . . . 9
⊢ ran
(𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)} |
| 52 | 40 | simplbi 497 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ⊆ 𝑋) |
| 53 | 52 | ad2antrl 728 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ⊆ 𝑋) |
| 54 | | ssrexv 4053 |
. . . . . . . . . . 11
⊢ (𝑣 ⊆ 𝑋 → (∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) |
| 55 | 53, 54 | 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))) |
| 56 | 55 | ss2abdv 4066 |
. . . . . . . . 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)}) |
| 57 | 51, 56 | eqsstrid 4022 |
. . . . . . . 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)}) |
| 58 | | unieq 4918 |
. . . . . . . . . . 11
⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑤 = ∪
ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑))) |
| 59 | 58 | eqeq1d 2739 |
. . . . . . . . . 10
⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (∪ 𝑤 = 𝑋 ↔ ∪ ran
(𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋)) |
| 60 | | ssabral 4065 |
. . . . . . . . . . 11
⊢ (𝑤 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) |
| 61 | | sseq1 4009 |
. . . . . . . . . . 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)})) |
| 62 | 60, 61 | bitr3id 285 |
. . . . . . . . . 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)})) |
| 63 | 59, 62 | anbi12d 632 |
. . . . . . . . 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)}))) |
| 64 | 63 | rspcev 3622 |
. . . . . . . 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))) |
| 65 | 45, 49, 57, 64 | syl12anc 837 |
. . . . . . 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))) |
| 66 | 65 | expr 456 |
. . . . . 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)))) |
| 67 | 66 | rexlimdva 3155 |
. . . . 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)))) |
| 68 | 39, 67 | impbid 212 |
. . . 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)) |
| 69 | 68 | ralbidv 3178 |
. . 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)) |
| 70 | 69 | pm5.32i 574 |
. 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)) |
| 71 | 1, 70 | bitri 275 |
1
⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) |