Theorem sstotbnd 38313

Description: Condition for a subset of a metric space to be totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Hypothesis

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

Assertion

Ref	Expression
sstotbnd	⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))

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

Allowed substitution hint:   𝑁(𝑏)

Proof of Theorem sstotbnd

Dummy variables 𝑓 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstotbnd.2	. . 3 ⊢ 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
2	1	sstotbnd2 38312	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
3		elfpw 9310	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢 ⊆ 𝑋 ∧ 𝑢 ∈ Fin))
4	3	simprbi 502	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5		mptfi 9307	. . . . . . . 8 ⊢ (𝑢 ∈ Fin → (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6		rnfi 9296	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin → ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
7	4, 5, 6	3syl 19	. . . . . . 7 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
8	7	ad2antrl 740	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))) → ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
9		simprr 784	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))) → 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))
10		eqid 2769	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
11	10	rnmpt 5948	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
12	3	simplbi 501	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ⊆ 𝑋)
13		ssrexv 4015	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑋 → (∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
14	12, 13	syl 18	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → (∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
15	14	ad2antrl 740	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))) → (∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
16	15	ss2abdv 4027	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))) → {𝑏 ∣ ∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)} ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})
17	11, 16	eqsstrid 3983	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))) → ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})
18		unieq 4887	. . . . . . . . . 10 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑣 = ∪ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)))
19		ovex 7444	. . . . . . . . . . 11 ⊢ (𝑥(ball‘𝑀)𝑑) ∈ V
20	19	dfiun3 5961	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) = ∪ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
21	18, 20	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑣 = ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))
22	21	sseq2d 3977	. . . . . . . 8 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 ⊆ ∪ 𝑣 ↔ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
23		ssabral 4026	. . . . . . . . 9 ⊢ (𝑣 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24		sseq1 3970	. . . . . . . . 9 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑣 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))
25	23, 24	bitr3id 288	. . . . . . . 8 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))
26	22, 25	anbi12d 643	. . . . . . 7 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ((𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) ∧ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})))
27	26	rspcev 3590	. . . . . 6 ⊢ ((ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) ∧ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})) → ∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
28	8, 9, 17, 27	syl12anc 849	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))) → ∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
29	28	rexlimdvaa 3173	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
30		oveq1 7418	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓‘𝑏)(ball‘𝑀)𝑑))
31	30	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
32	31	ac6sfi 9243	. . . . . . . 8 ⊢ ((𝑣 ∈ Fin ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
33	32	adantrl 728	. . . . . . 7 ⊢ ((𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → ∃𝑓(𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
34	33	adantl 486	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → ∃𝑓(𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
35		frn 6714	. . . . . . . . 9 ⊢ (𝑓:𝑣⟶𝑋 → ran 𝑓 ⊆ 𝑋)
36	35	ad2antrl 740	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → ran 𝑓 ⊆ 𝑋)
37		simplrl 788	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → 𝑣 ∈ Fin)
38		ffn 6706	. . . . . . . . . . 11 ⊢ (𝑓:𝑣⟶𝑋 → 𝑓 Fn 𝑣)
39	38	ad2antrl 740	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → 𝑓 Fn 𝑣)
40		dffn4 6799	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑣 ↔ 𝑓:𝑣–onto→ran 𝑓)
41	39, 40	sylib 221	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → 𝑓:𝑣–onto→ran 𝑓)
42		fofi 9272	. . . . . . . . 9 ⊢ ((𝑣 ∈ Fin ∧ 𝑓:𝑣–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
43	37, 41, 42	syl2anc 595	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → ran 𝑓 ∈ Fin)
44		elfpw 9310	. . . . . . . 8 ⊢ (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑋 ∧ ran 𝑓 ∈ Fin))
45	36, 43, 44	sylanbrc 594	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin))
46		simprrl 792	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → 𝑌 ⊆ ∪ 𝑣)
47	46	adantr 485	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → 𝑌 ⊆ ∪ 𝑣)
48		uniiun 5027	. . . . . . . . . . 11 ⊢ ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 𝑏
49		iuneq2 4980	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑏 ∈ 𝑣 𝑏 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
50	48, 49	eqtrid 2816	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
51	50	ad2antll 741	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
52	47, 51	sseqtrd 3981	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → 𝑌 ⊆ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
53	30	eleq2d 2855	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
54	53	rexrn 7083	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑣 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
55		eliun 4964	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56		eliun 4964	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑣 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))
57	54, 55, 56	3bitr4g 317	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑣 → (𝑦 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
58	57	eqrdv 2767	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑣 → ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
59	39, 58	syl 18	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
60	52, 59	sseqtrrd 3982	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → 𝑌 ⊆ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
61		iuneq1 4977	. . . . . . . . 9 ⊢ (𝑢 = ran 𝑓 → ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
62	61	sseq2d 3977	. . . . . . . 8 ⊢ (𝑢 = ran 𝑓 → (𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 ⊆ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
63	62	rspcev 3590	. . . . . . 7 ⊢ ((ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))
64	45, 60, 63	syl2anc 595	. . . . . 6 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) ∧ (𝑓:𝑣⟶𝑋 ∧ ∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))) → ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))
65	34, 64	exlimddv 1962	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))
66	65	rexlimdvaa 3173	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
67	29, 66	impbid 215	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
68	67	ralbidv 3194	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∀𝑑 ∈ ℝ+ ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
69	2, 68	bitrd 282	1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876  ∪ ciun 4960   ↦ cmpt 5196   × cxp 5660  ran crn 5663   ↾ cres 5664   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7411  Fincfn 8942  ℝ⁺crp 13015  Metcmet 21476  ballcbl 21477  TotBndctotbnd 38304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-psmet 21482  df-xmet 21483  df-met 21484  df-bl 21485  df-totbnd 38306

This theorem is referenced by:  totbndss  38315