Theorem sstotbnd 32744

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 32743	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑢 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
3		elfpw 8151	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑢 ⊆ 𝑋 ∧ 𝑢 ∈ Fin))
4	3	simprbi 479	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ∈ Fin)
5		mptfi 8148	. . . . . . . 8 ⊢ (𝑢 ∈ Fin → (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
6		rnfi 8132	. . . . . . . 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 760	. . . . . 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 792	. . . . . 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 2610	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
11	10	rnmpt 5292	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥 ∈ 𝑢 𝑏 = (𝑥(ball‘𝑀)𝑑)}
12	3	simplbi 475	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝒫 𝑋 ∩ Fin) → 𝑢 ⊆ 𝑋)
13		ssrexv 3630	. . . . . . . . . 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 760	. . . . . . . 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 3638	. . . . . . 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	syl5eqss 3612	. . . . . 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 4380	. . . . . . . . . 10 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑣 = ∪ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)))
19		ovex 6577	. . . . . . . . . . 11 ⊢ (𝑥(ball‘𝑀)𝑑) ∈ V
20	19	dfiun3 5301	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) = ∪ ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑))
21	18, 20	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑣 = ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑))
22	21	sseq2d 3596	. . . . . . . 8 ⊢ (𝑣 = ran (𝑥 ∈ 𝑢 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑌 ⊆ ∪ 𝑣 ↔ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑)))
23		ssabral 3636	. . . . . . . . 9 ⊢ (𝑣 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
24		sseq1 3589	. . . . . . . . 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 273	. . . . . . . 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 743	. . . . . . 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 3282	. . . . . 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 1316	. . . . 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 3014	. . . 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 6556	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓‘𝑏)(ball‘𝑀)𝑑))
31	30	eqeq2d 2620	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
32	31	ac6sfi 8089	. . . . . . . 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 748	. . . . . . 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 5966	. . . . . . . . 9 ⊢ (𝑓:𝑣⟶𝑋 → ran 𝑓 ⊆ 𝑋)
36	35	ad2antrl 760	. . . . . . . 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 796	. . . . . . . . 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 5958	. . . . . . . . . . 11 ⊢ (𝑓:𝑣⟶𝑋 → 𝑓 Fn 𝑣)
39	38	ad2antrl 760	. . . . . . . . . 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 6034	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑣 ↔ 𝑓:𝑣–onto→ran 𝑓)
41	39, 40	sylib 207	. . . . . . . . 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 8135	. . . . . . . . 9 ⊢ ((𝑣 ∈ Fin ∧ 𝑓:𝑣–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
43	37, 41, 42	syl2anc 691	. . . . . . . 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 8151	. . . . . . . 8 ⊢ (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑋 ∧ ran 𝑓 ∈ Fin))
45	36, 43, 44	sylanbrc 695	. . . . . . 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 800	. . . . . . . . . 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 4509	. . . . . . . . . . 11 ⊢ ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 𝑏
49		iuneq2 4473	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑏 ∈ 𝑣 𝑏 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
50	48, 49	syl5eq 2656	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑣 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑣 = ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
51	50	ad2antll 761	. . . . . . . . 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 3604	. . . . . . . 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 2673	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
54	53	rexrn 6269	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑣 → (∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑣 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
55		eliun 4460	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑦 ∈ (𝑥(ball‘𝑀)𝑑))
56		eliun 4460	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑣 𝑦 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))
57	54, 55, 56	3bitr4g 302	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑣 → (𝑦 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑦 ∈ ∪ 𝑏 ∈ 𝑣 ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
58	57	eqrdv 2608	. . . . . . . . 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 3605	. . . . . . 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 4470	. . . . . . . . 9 ⊢ (𝑢 = ran 𝑓 → ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
62	61	sseq2d 3596	. . . . . . . 8 ⊢ (𝑢 = ran 𝑓 → (𝑌 ⊆ ∪ 𝑥 ∈ 𝑢 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 ⊆ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑)))
63	62	rspcev 3282	. . . . . . 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 691	. . . . . 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 1850	. . . . 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 3014	. . . 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 201	. . 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 2969	. 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 267	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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℝ⁺crp 11708  Metcme 19553  ballcbl 19554  TotBndctotbnd 32735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-2 10956  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-totbnd 32737

This theorem is referenced by:  totbndss  32746