Theorem sstotbnd3 37763

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 37761	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)))
3		elin 3927	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin))
4		rabfi 9190	. . . . . . . . . 10 ⊢ (𝑣 ∈ Fin → {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)
5	4	anim2i 617	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
6	3, 5	sylbi 217	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
7	6	anim2i 617	. . . . . . 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 458	. . . . . 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 645	. . . . . 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 218	. . . . 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 3062	. . . 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 3066	. . 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 253	. 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 4039	. . . . . . . 8 ⊢ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑣
15		elpwi 4566	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝒫 𝑋 → 𝑣 ⊆ 𝑋)
16	15	ad2antlr 727	. . . . . . . 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 3955	. . . . . . 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 772	. . . . . . 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 9281	. . . . . . 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 583	. . . . . 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 3938	. . . . . . . . . . . 12 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑))
22		eliun 4955	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ 𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
23	21, 22	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
24		inelcm 4424	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅)
25	24	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅))
26	25	ancrd 551	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))))
27	26	reximdv 3148	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 588	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
30		eliun 4955	. . . . . . . . . . 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 7376	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)𝑑))
32	31	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (𝑦(ball‘𝑀)𝑑) ↔ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
33	32	rexrab2 3668	. . . . . . . . . . 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 275	. . . . . . . . . 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 234	. . . . . . . . 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 412	. . . . . . . 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 3949	. . . . . . 7 ⊢ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → 𝑌 ⊆ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
38	37	ad2antrl 728	. . . . . 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 4968	. . . . . . . 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 3976	. . . . . . 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 3585	. . . . . 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 584	. . . . 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 3136	. . . 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 3147	. . 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 37761	. . 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 259	. 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 212	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3402   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  ∪ ciun 4951   × cxp 5629   ↾ cres 5633  ‘cfv 6499  (class class class)co 7369  Fincfn 8895  ℝ⁺crp 12927  Metcmet 21282  ballcbl 21283  TotBndctotbnd 37753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-rp 12928  df-xneg 13048  df-xadd 13049  df-xmul 13050  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-totbnd 37755

This theorem is referenced by:  cntotbnd  37783