Theorem sstotbnd3 38529

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 38527	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)))
3		elin 3915	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin))
4		rabfi 9244	. . . . . . . . . 10 ⊢ (𝑣 ∈ Fin → {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)
5	4	anim2i 629	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
6	3, 5	sylbi 220	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
7	6	anim2i 629	. . . . . . 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 464	. . . . . 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 658	. . . . . 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 221	. . . . 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 3095	. . . 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 3099	. . 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 256	. 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 4028	. . . . . . . 8 ⊢ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑣
15		elpwi 4564	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝒫 𝑋 → 𝑣 ⊆ 𝑋)
16	15	ad2antlr 740	. . . . . . . 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 3942	. . . . . . 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 785	. . . . . . 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 9324	. . . . . . 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 595	. . . . . 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 3926	. . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ∃𝑥 ∈ 𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
24		inelcm 4418	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (𝑥(ball‘𝑀)𝑑) ∧ 𝑧 ∈ 𝑌) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅)
25	24	expcom 419	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅))
26	25	ancrd 561	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))))
27	26	reximdv 3177	. . . . . . . . . . . 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 413	. . . . . . . . . . 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 600	. . . . . . . . . 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 7421	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)𝑑))
32	31	eleq2d 2846	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑧 ∈ (𝑦(ball‘𝑀)𝑑) ↔ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
33	32	rexrab2 3658	. . . . . . . . . . 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 278	. . . . . . . . . 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 237	. . . . . . . . 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 418	. . . . . . . 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 3937	. . . . . . 7 ⊢ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → 𝑌 ⊆ ∪ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
38	37	ad2antrl 741	. . . . . 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 3963	. . . . . . 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 3576	. . . . . 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 596	. . . . 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 3165	. . . 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 3176	. . 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 38527	. . 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 262	. 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 215	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ ciun 4951   × cxp 5653   ↾ cres 5657  ‘cfv 6533  (class class class)co 7414  Fincfn 8955  ℝ⁺crp 13045  Metcmet 21574  ballcbl 21575  TotBndctotbnd 38519

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-rp 13046  df-xneg 13166  df-xadd 13167  df-xmul 13168  df-psmet 21580  df-xmet 21581  df-met 21582  df-bl 21583  df-totbnd 38521

This theorem is used by:  cntotbnd  38549