Theorem sstotbnd 37823

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 37822	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
3		elfpw 9238	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢 ⊆ 𝑋 ∧ 𝑢 ∈ Fin))
4	3	simprbi 496	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5		mptfi 9235	. . . . . . . 8 ⊢ (𝑢 ∈ Fin → (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6		rnfi 9224	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin → ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
7	4, 5, 6	3syl 18	. . . . . . 7 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
8	7	ad2antrl 728	. . . . . 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 772	. . . . . 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 2731	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
11	10	rnmpt 5896	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
12	3	simplbi 497	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ⊆ 𝑋)
13		ssrexv 3999	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑋 → (∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → (∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
15	14	ad2antrl 728	. . . . . . . 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 4012	. . . . . . 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 3968	. . . . . 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 4867	. . . . . . . . . 10 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑣 = ∪ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)))
19		ovex 7379	. . . . . . . . . . 11 ⊢ (𝑥(ball‘𝑀)𝑑) ∈ V
20	19	dfiun3 5908	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) = ∪ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
21	18, 20	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑣 = ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))
22	21	sseq2d 3962	. . . . . . . 8 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 ⊆ ∪ 𝑣 ↔ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
23		ssabral 4011	. . . . . . . . 9 ⊢ (𝑣 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24		sseq1 3955	. . . . . . . . 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 285	. . . . . . . 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 632	. . . . . . 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 3572	. . . . . 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 836	. . . . 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 3134	. . . 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 7353	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓‘𝑏)(ball‘𝑀)𝑑))
31	30	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
32	31	ac6sfi 9168	. . . . . . . 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 716	. . . . . . 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 481	. . . . . 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 6658	. . . . . . . . 9 ⊢ (𝑓:𝑣⟶𝑋 → ran 𝑓 ⊆ 𝑋)
36	35	ad2antrl 728	. . . . . . . 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 776	. . . . . . . . 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 6651	. . . . . . . . . . 11 ⊢ (𝑓:𝑣⟶𝑋 → 𝑓 Fn 𝑣)
39	38	ad2antrl 728	. . . . . . . . . 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 6741	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑣 ↔ 𝑓:𝑣–onto→ran 𝑓)
41	39, 40	sylib 218	. . . . . . . . 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 9197	. . . . . . . . 9 ⊢ ((𝑣 ∈ Fin ∧ 𝑓:𝑣–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
43	37, 41, 42	syl2anc 584	. . . . . . . 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 9238	. . . . . . . 8 ⊢ (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑋 ∧ ran 𝑓 ∈ Fin))
45	36, 43, 44	sylanbrc 583	. . . . . . 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 780	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → 𝑌 ⊆ ∪ 𝑣)
47	46	adantr 480	. . . . . . . . 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 5005	. . . . . . . . . . 11 ⊢ ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 𝑏
49		iuneq2 4959	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑏 ∈ 𝑣 𝑏 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
50	48, 49	eqtrid 2778	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
51	50	ad2antll 729	. . . . . . . . 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 3966	. . . . . . . 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 2817	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
54	53	rexrn 7020	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑣 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
55		eliun 4943	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56		eliun 4943	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑣 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))
57	54, 55, 56	3bitr4g 314	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑣 → (𝑦 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
58	57	eqrdv 2729	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑣 → ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
59	39, 58	syl 17	. . . . . . . 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 3967	. . . . . . 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 4956	. . . . . . . . 9 ⊢ (𝑢 = ran 𝑓 → ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
62	61	sseq2d 3962	. . . . . . . 8 ⊢ (𝑢 = ran 𝑓 → (𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 ⊆ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
63	62	rspcev 3572	. . . . . . 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 584	. . . . . 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 1936	. . . . 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 3134	. . . 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 212	. . 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 3155	. 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 279	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 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856  ∪ ciun 4939   ↦ cmpt 5170   × cxp 5612  ran crn 5615   ↾ cres 5616   Fn wfn 6476  ⟶wf 6477  –onto→wfo 6479  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  ℝ⁺crp 12890  Metcmet 21277  ballcbl 21278  TotBndctotbnd 37814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-rp 12891  df-xneg 13011  df-xadd 13012  df-xmul 13013  df-psmet 21283  df-xmet 21284  df-met 21285  df-bl 21286  df-totbnd 37816

This theorem is referenced by:  totbndss  37825