Theorem istotbnd3 33570

Description: A metric space is totally bounded iff there is a finite ε-net for every positive ε. This differs from the definition in providing a finite set of ball centers rather than a finite set of balls. (Contributed by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
istotbnd3	⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))

Distinct variable groups:   𝑣,𝑑,𝑥,𝑀   𝑋,𝑑,𝑣,𝑥

Proof of Theorem istotbnd3

Dummy variables 𝑏 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		istotbnd 33568	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2		oveq1 6657	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓‘𝑏)(ball‘𝑀)𝑑))
3	2	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
4	3	ac6sfi 8204	. . . . . . . . . 10 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
5	4	ex 450	. . . . . . . . 9 ⊢ (𝑤 ∈ Fin → (∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑓(𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))))
6	5	ad2antlr 763	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ∪ 𝑤 = 𝑋) → (∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑓(𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))))
7		simprrl 804	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤⟶𝑋)
8		frn 6053	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑤⟶𝑋 → ran 𝑓 ⊆ 𝑋)
9	7, 8	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ⊆ 𝑋)
10		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑤 ∈ Fin)
11		ffn 6045	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑤⟶𝑋 → 𝑓 Fn 𝑤)
12	7, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓 Fn 𝑤)
13		dffn4 6121	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝑤 ↔ 𝑓:𝑤–onto→ran 𝑓)
14	12, 13	sylib 208	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤–onto→ran 𝑓)
15		fofi 8252	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Fin ∧ 𝑓:𝑤–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
16	10, 14, 15	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ Fin)
17		elfpw 8268	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑋 ∧ ran 𝑓 ∈ Fin))
18	9, 16, 17	sylanbrc 698	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin))
19	2	eleq2d 2687	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑓‘𝑏) → (𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
20	19	rexrn 6361	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn 𝑤 → (∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑤 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
21	12, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → (∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑤 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
22		eliun 4524	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑))
23		eliun 4524	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏 ∈ 𝑤 𝑣 ∈ ((𝑓‘𝑏)(ball‘𝑀)𝑑))
24	21, 22, 23	3bitr4g 303	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → (𝑣 ∈ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑣 ∈ ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑)))
25	24	eqrdv 2620	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
26		simprrr 805	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑))
27		iuneq2 4537	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑) → ∪ 𝑏 ∈ 𝑤 𝑏 = ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
28	26, 27	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑏 ∈ 𝑤 𝑏 = ∪ 𝑏 ∈ 𝑤 ((𝑓‘𝑏)(ball‘𝑀)𝑑))
29		uniiun 4573	. . . . . . . . . . . . 13 ⊢ ∪ 𝑤 = ∪ 𝑏 ∈ 𝑤 𝑏
30		simprl 794	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑤 = 𝑋)
31	29, 30	syl5eqr 2670	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑏 ∈ 𝑤 𝑏 = 𝑋)
32	25, 28, 31	3eqtr2d 2662	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋)
33		iuneq1 4534	. . . . . . . . . . . . 13 ⊢ (𝑣 = ran 𝑓 → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
34	33	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑣 = ran 𝑓 → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋))
35	34	rspcev 3309	. . . . . . . . . . 11 ⊢ ((ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
36	18, 32, 35	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ (∪ 𝑤 = 𝑋 ∧ (𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)))) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
37	36	expr 643	. . . . . . . . 9 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ∪ 𝑤 = 𝑋) → ((𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
38	37	exlimdv 1861	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ∪ 𝑤 = 𝑋) → (∃𝑓(𝑓:𝑤⟶𝑋 ∧ ∀𝑏 ∈ 𝑤 𝑏 = ((𝑓‘𝑏)(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
39	6, 38	syld 47	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ∪ 𝑤 = 𝑋) → (∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
40	39	expimpd 629	. . . . . 6 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) → ((∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
41	40	rexlimdva 3031	. . . . 5 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
42		elfpw 8268	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ⊆ 𝑋 ∧ 𝑣 ∈ Fin))
43	42	simprbi 480	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
44	43	ad2antrl 764	. . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ∈ Fin)
45		mptfi 8265	. . . . . . . . 9 ⊢ (𝑣 ∈ Fin → (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
46		rnfi 8249	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin → ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
47	44, 45, 46	3syl 18	. . . . . . . 8 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
48		ovex 6678	. . . . . . . . . 10 ⊢ (𝑥(ball‘𝑀)𝑑) ∈ V
49	48	dfiun3 5380	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑))
50		simprr 796	. . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
51	49, 50	syl5eqr 2670	. . . . . . . 8 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋)
52		eqid 2622	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑))
53	52	rnmpt 5371	. . . . . . . . 9 ⊢ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)}
54	42	simplbi 476	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ⊆ 𝑋)
55	54	ad2antrl 764	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ⊆ 𝑋)
56		ssrexv 3667	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝑋 → (∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
57	55, 56	syl 17	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → (∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
58	57	ss2abdv 3675	. . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → {𝑏 ∣ ∃𝑥 ∈ 𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)} ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})
59	53, 58	syl5eqss 3649	. . . . . . . 8 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})
60		unieq 4444	. . . . . . . . . . 11 ⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → ∪ 𝑤 = ∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)))
61	60	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (∪ 𝑤 = 𝑋 ↔ ∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋))
62		ssabral 3673	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
63		sseq1 3626	. . . . . . . . . . 11 ⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑤 ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))
64	62, 63	syl5bbr 274	. . . . . . . . . 10 ⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))
65	61, 64	anbi12d 747	. . . . . . . . 9 ⊢ (𝑤 = ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → ((∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ (∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋 ∧ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})))
66	65	rspcev 3309	. . . . . . . 8 ⊢ ((ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin ∧ (∪ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋 ∧ ran (𝑥 ∈ 𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})) → ∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
67	47, 51, 59, 66	syl12anc 1324	. . . . . . 7 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
68	67	expr 643	. . . . . 6 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
69	68	rexlimdva 3031	. . . . 5 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
70	41, 69	impbid 202	. . . 4 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
71	70	ralbidv 2986	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → (∀𝑑 ∈ ℝ+ ∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
72	71	pm5.32i 669	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑤 ∈ Fin (∪ 𝑤 = 𝑋 ∧ ∀𝑏 ∈ 𝑤 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
73	1, 72	bitri 264	1 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520   ↦ cmpt 4729  ran crn 5115   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℝ⁺crp 11832  Metcme 19732  ballcbl 19733  TotBndctotbnd 33565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-totbnd 33567

This theorem is referenced by:  0totbnd  33572  sstotbnd2  33573  equivtotbnd  33577  totbndbnd  33588  prdstotbnd  33593