Theorem sstotbnd3 38158

Description: Use a net that is not necessarily finite, but for which only finitely many balls meet the subset. (Contributed by Mario Carneiro, 14-Sep-2015.)

Hypothesis

Ref	Expression
sstotbnd.2	⊢ 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))

Assertion

Ref	Expression
sstotbnd3	⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))

Distinct variable groups:   𝑣,𝑑,𝑥,𝑀   𝑋,𝑑,𝑣,𝑥   𝑁,𝑑,𝑣,𝑥   𝑌,𝑑,𝑣,𝑥

Proof of Theorem sstotbnd3

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstotbnd.2	. . . 4 ⊢ 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
2	1	sstotbnd2 38156	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)))
3		elin 3901	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin))
4		rabfi 9175	. . . . . . . . . 10 ⊢ (𝑣 ∈ Fin → {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)
5	4	anim2i 624	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
6	3, 5	sylbi 219	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
7	6	anim2i 624	. . . . . . 7 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
8	7	ancoms 460	. . . . . 6 ⊢ ((𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)) → (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
9		an12 652	. . . . . 6 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
10	8, 9	sylib 220	. . . . 5 ⊢ ((𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)) → (𝑣 ∈ 𝒫 𝑋 ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
11	10	reximi2 3074	. . . 4 ⊢ (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
12	11	ralimi 3078	. . 3 ⊢ (∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
13	2, 12	biimtrdi 255	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) → ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
14		ssrab2 4014	. . . . . . . 8 ⊢ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑣
15		elpwi 4539	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝒫 𝑋 → 𝑣 ⊆ 𝑋)
16	15	ad2antlr 734	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → 𝑣 ⊆ 𝑋)
17	14, 16	sstrid 3928	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑋)
18		simprr 779	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)
19		elfpw 9258	. . . . . . 7 ⊢ ({𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin) ↔ ({𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
20	17, 18, 19	sylanbrc 590	. . . . . 6 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin))
21		ssel2 3912	. . . . . . . . . . . 12 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑))
22		eliun 4928	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ 𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
23	21, 22	sylib 220	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
24		inelcm 4396	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅)
25	24	expcom 415	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅))
26	25	ancrd 557	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))))
27	26	reximdv 3156	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑌 → (∃𝑥 ∈ 𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))))
28	27	impcom 409	. . . . . . . . . . 11 ⊢ ((∃𝑥 ∈ 𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
29	23, 28	sylancom 595	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
30		eliun 4928	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅}𝑧 ∈ (𝑦(ball‘𝑀)𝑑))
31		oveq1 7367	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)𝑑))
32	31	eleq2d 2827	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (𝑦(ball‘𝑀)𝑑) ↔ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
33	32	rexrab2 3643	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅}𝑧 ∈ (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
34	30, 33	bitri 277	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
35	29, 34	sylibr 236	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
36	35	ex 414	. . . . . . . 8 ⊢ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → (𝑧 ∈ 𝑌 → 𝑧 ∈ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)))
37	36	ssrdv 3923	. . . . . . 7 ⊢ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → 𝑌 ⊆ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
38	37	ad2antrl 735	. . . . . 6 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → 𝑌 ⊆ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
39		iuneq1 4941	. . . . . . . 8 ⊢ (𝑤 = {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} → ∪ 𝑦 ∈ 𝑤 (𝑦(ball‘𝑀)𝑑) = ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
40	39	sseq2d 3949	. . . . . . 7 ⊢ (𝑤 = {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} → (𝑌 ⊆ ∪ 𝑦 ∈ 𝑤 (𝑦(ball‘𝑀)𝑑) ↔ 𝑌 ⊆ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)))
41	40	rspcev 3562	. . . . . 6 ⊢ (({𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑦 ∈ 𝑤 (𝑦(ball‘𝑀)𝑑))
42	20, 38, 41	syl2anc 591	. . . . 5 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑦 ∈ 𝑤 (𝑦(ball‘𝑀)𝑑))
43	42	rexlimdva2 3144	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑦 ∈ 𝑤 (𝑦(ball‘𝑀)𝑑)))
44	43	ralimdv 3155	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∀𝑑 ∈ ℝ+ ∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → ∀𝑑 ∈ ℝ+ ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑦 ∈ 𝑤 (𝑦(ball‘𝑀)𝑑)))
45	1	sstotbnd2 38156	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑦 ∈ 𝑤 (𝑦(ball‘𝑀)𝑑)))
46	44, 45	sylibrd 261	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∀𝑑 ∈ ℝ+ ∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → 𝑁 ∈ (TotBnd‘𝑌)))
47	13, 46	impbid 214	1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  {crab 3393   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  𝒫 cpw 4532  ∪ ciun 4924   × cxp 5619   ↾ cres 5623  ‘cfv 6489  (class class class)co 7360  Fincfn 8887  ℝ⁺crp 12937  Metcmet 21337  ballcbl 21338  TotBndctotbnd 38148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-psmet 21343  df-xmet 21344  df-met 21345  df-bl 21346  df-totbnd 38150

This theorem is referenced by:  cntotbnd  38178