Theorem sstotbnd 34604

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 34603	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
3		elfpw 8672	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢 ⊆ 𝑋 ∧ 𝑢 ∈ Fin))
4	3	simprbi 497	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5		mptfi 8669	. . . . . . . 8 ⊢ (𝑢 ∈ Fin → (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6		rnfi 8653	. . . . . . . 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 724	. . . . . 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 769	. . . . . 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 2795	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
11	10	rnmpt 5709	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
12	3	simplbi 498	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ⊆ 𝑋)
13		ssrexv 3955	. . . . . . . . . 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 724	. . . . . . . 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 3965	. . . . . . 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 3936	. . . . . 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 4753	. . . . . . . . . 10 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑣 = ∪ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)))
19		ovex 7048	. . . . . . . . . . 11 ⊢ (𝑥(ball‘𝑀)𝑑) ∈ V
20	19	dfiun3 5718	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) = ∪ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
21	18, 20	syl6eqr 2849	. . . . . . . . 9 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑣 = ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))
22	21	sseq2d 3920	. . . . . . . 8 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 ⊆ ∪ 𝑣 ↔ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
23		ssabral 3963	. . . . . . . . 9 ⊢ (𝑣 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24		sseq1 3913	. . . . . . . . 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	syl5bbr 286	. . . . . . . 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 630	. . . . . . 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 3559	. . . . . 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 833	. . . . 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 3248	. . . 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 7023	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓‘𝑏)(ball‘𝑀)𝑑))
31	30	eqeq2d 2805	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
32	31	ac6sfi 8608	. . . . . . . 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 712	. . . . . . 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 482	. . . . . 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 6388	. . . . . . . . 9 ⊢ (𝑓:𝑣⟶𝑋 → ran 𝑓 ⊆ 𝑋)
36	35	ad2antrl 724	. . . . . . . 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 773	. . . . . . . . 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 6382	. . . . . . . . . . 11 ⊢ (𝑓:𝑣⟶𝑋 → 𝑓 Fn 𝑣)
39	38	ad2antrl 724	. . . . . . . . . 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 6464	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑣 ↔ 𝑓:𝑣–onto→ran 𝑓)
41	39, 40	sylib 219	. . . . . . . . 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 8656	. . . . . . . . 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 8672	. . . . . . . 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 777	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑣 ∈ Fin ∧ (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) → 𝑌 ⊆ ∪ 𝑣)
47	46	adantr 481	. . . . . . . . 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 4881	. . . . . . . . . . 11 ⊢ ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 𝑏
49		iuneq2 4843	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑏 ∈ 𝑣 𝑏 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
50	48, 49	syl5eq 2843	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
51	50	ad2antll 725	. . . . . . . . 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 3928	. . . . . . . 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 2868	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
54	53	rexrn 6718	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑣 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
55		eliun 4829	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56		eliun 4829	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑣 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))
57	54, 55, 56	3bitr4g 315	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑣 → (𝑦 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
58	57	eqrdv 2793	. . . . . . . . 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	sseqtr4d 3929	. . . . . . 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 4840	. . . . . . . . 9 ⊢ (𝑢 = ran 𝑓 → ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
62	61	sseq2d 3920	. . . . . . . 8 ⊢ (𝑢 = ran 𝑓 → (𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 ⊆ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
63	62	rspcev 3559	. . . . . . 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 1913	. . . . 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 3248	. . . 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 213	. . 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 3164	. 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 280	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 207   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081  {cab 2775  ∀wral 3105  ∃wrex 3106   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4453  ∪ cuni 4745  ∪ ciun 4825   ↦ cmpt 5041   × cxp 5441  ran crn 5444   ↾ cres 5445   Fn wfn 6220  ⟶wf 6221  –onto→wfo 6223  ‘cfv 6225  (class class class)co 7016  Fincfn 8357  ℝ⁺crp 12239  Metcmet 20213  ballcbl 20214  TotBndctotbnd 34595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-2 11548  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-totbnd 34597

This theorem is referenced by:  totbndss  34606