Theorem heibor1 38489

Description: One half of heibor 38500, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 25489 and total boundedness here, which follows trivially from the fact that the set of all 𝑟-balls is an open cover of 𝑋, so finitely many cover 𝑋. (Contributed by Jeff Madsen, 16-Jan-2014.)

Hypothesis

Ref	Expression
heibor.1	⊢ 𝐽 = (MetOpen‘𝐷)

Assertion

Ref	Expression
heibor1	⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Proof of Theorem heibor1

Dummy variables 𝑥 𝑦 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		heibor.1	. . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷)
2		simpll 778	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐷 ∈ (Met‘𝑋))
3		simplr 780	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐽 ∈ Comp)
4		simprl 782	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ (Cau‘𝐷))
5		simprr 784	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥:ℕ⟶𝑋)
6	1, 2, 3, 4, 5	heibor1lem 38488	. . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ dom (⇝𝑡‘𝐽))
7	6	expr 461	. . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑥 ∈ (Cau‘𝐷)) → (𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
8	7	ralrimiva 3156	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
9		nnuz 12907	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10		1zzd 12631	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 1 ∈ ℤ)
11		simpl 487	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (Met‘𝑋))
12	9, 1, 10, 11	iscmet3 25463	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽))))
13	8, 12	mpbird 260	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (CMet‘𝑋))
14		simplr 780	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐽 ∈ Comp)
15		metxmet 24502	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
16		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → 𝑧 ∈ 𝑋)
17		rpxr 13032	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
18	1	blopn 24668	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
19	15, 16, 17, 18	syl3an 1177	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
20	19	3com23 1143	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
21	20	3expa 1135	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
22		eleq1a 2857	. . . . . . . . . . 11 ⊢ ((𝑧(ball‘𝐷)𝑟) ∈ 𝐽 → (𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
23	21, 22	syl 18	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
24	23	rexlimdva 3165	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
25	24	adantlr 727	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
26	25	abssdv 4020	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ 𝐽)
27	15	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑋))
28	1	mopnuni 24609	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	27, 28	syl 18	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 = ∪ 𝐽)
30		blcntr 24581	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
31	15, 30	syl3an1 1180	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
32	31	3com23 1143	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
33	32	3expa 1135	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
34		ovex 7445	. . . . . . . . . . . . . . 15 ⊢ (𝑧(ball‘𝐷)𝑟) ∈ V
35	34	elabrex 7240	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → (𝑧(ball‘𝐷)𝑟) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
36	35	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
37		elunii 4876	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝑧(ball‘𝐷)𝑟) ∧ (𝑧(ball‘𝐷)𝑟) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}) → 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
38	33, 36, 37	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
39	38	ralrimiva 3156	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
40	39	adantlr 727	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
41		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝑋
42		nfre1 3289	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)
43	42	nfab 2930	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧{𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
44	43	nfuni 4878	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
45	41, 44	dfss3f 3928	. . . . . . . . . 10 ⊢ (𝑋 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
46	40, 45	sylibr 237	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
47	29, 46	eqsstrrd 3971	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
48	26	unissd 4881	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ ∪ 𝐽)
49	47, 48	eqssd 3953	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
50		eqid 2762	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
51	50	cmpcov 23557	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}) → ∃𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin)∪ 𝐽 = ∪ 𝑥)
52	14, 26, 49, 51	syl3anc 1397	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin)∪ 𝐽 = ∪ 𝑥)
53		elin 3920	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin) ↔ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ 𝑥 ∈ Fin))
54		ancom 465	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ 𝑥 ∈ Fin) ↔ (𝑥 ∈ Fin ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}))
55	53, 54	bitri 278	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin) ↔ (𝑥 ∈ Fin ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}))
56	55	anbi1i 635	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin) ∧ ∪ 𝐽 = ∪ 𝑥) ↔ ((𝑥 ∈ Fin ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}) ∧ ∪ 𝐽 = ∪ 𝑥))
57		anass 473	. . . . . . . 8 ⊢ (((𝑥 ∈ Fin ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}) ∧ ∪ 𝐽 = ∪ 𝑥) ↔ (𝑥 ∈ Fin ∧ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥)))
58	56, 57	bitri 278	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin) ∧ ∪ 𝐽 = ∪ 𝑥) ↔ (𝑥 ∈ Fin ∧ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥)))
59	58	rexbii2 3107	. . . . . 6 ⊢ (∃𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin)∪ 𝐽 = ∪ 𝑥 ↔ ∃𝑥 ∈ Fin (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥))
60	52, 59	sylib 221	. . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ Fin (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥))
61		ancom 465	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥) ↔ (∪ 𝐽 = ∪ 𝑥 ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}))
62		eqcom 2769	. . . . . . . . . 10 ⊢ (∪ 𝑥 = 𝑋 ↔ 𝑋 = ∪ 𝑥)
63	29	eqeq1d 2764	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (𝑋 = ∪ 𝑥 ↔ ∪ 𝐽 = ∪ 𝑥))
64	62, 63	bitr2id 287	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∪ 𝐽 = ∪ 𝑥 ↔ ∪ 𝑥 = 𝑋))
65	64	anbi1d 642	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ((∪ 𝐽 = ∪ 𝑥 ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}) ↔ (∪ 𝑥 = 𝑋 ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})))
66	61, 65	bitrid 286	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ((𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥) ↔ (∪ 𝑥 = 𝑋 ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})))
67		elpwi 4568	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → 𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
68		ssabral 4017	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))
69	67, 68	sylib 221	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))
70	69	anim2i 628	. . . . . . 7 ⊢ ((∪ 𝑥 = 𝑋 ∧ 𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}) → (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)))
71	66, 70	biimtrdi 256	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ((𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥) → (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))))
72	71	reximdv 3179	. . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∃𝑥 ∈ Fin (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥) → ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))))
73	60, 72	mpd 16	. . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)))
74	73	ralrimiva 3156	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑟 ∈ ℝ+ ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)))
75		istotbnd 38448	. . 3 ⊢ (𝐷 ∈ (TotBnd‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑟 ∈ ℝ+ ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))))
76	11, 74, 75	sylanbrc 594	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (TotBnd‘𝑋))
77	13, 76	jca 520	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  ∃wrex 3088   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871  dom cdm 5660  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412  Fincfn 8941  1c1 11107  ℝ^*cxr 11248  ℕcn 12239  ℝ⁺crp 13022  ∞Metcxmet 21518  Metcmet 21519  ballcbl 21520  MetOpencmopn 21523  ⇝_𝑡clm 23394  Compccmp 23554  Cauccau 25423  CMetccmet 25424  TotBndctotbnd 38445

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-inf2 9608  ax-cc 10425  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-omul 8456  df-er 8692  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9932  df-acn 9935  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-div 11878  df-nn 12240  df-2 12309  df-3 12310  df-n0 12511  df-z 12598  df-uz 12869  df-q 12979  df-rp 13023  df-xneg 13143  df-xadd 13144  df-xmul 13145  df-ico 13384  df-fz 13542  df-fl 13832  df-seq 14045  df-exp 14105  df-cj 15157  df-re 15158  df-im 15159  df-sqrt 15293  df-abs 15294  df-clim 15546  df-rlim 15547  df-rest 17481  df-topgen 17502  df-psmet 21525  df-xmet 21526  df-met 21527  df-bl 21528  df-mopn 21529  df-fbas 21530  df-fg 21531  df-top 23062  df-topon 23079  df-bases 23114  df-cld 23187  df-ntr 23188  df-cls 23189  df-nei 23266  df-lm 23397  df-cmp 23555  df-fil 24014  df-fm 24106  df-flim 24107  df-flf 24108  df-cfil 25425  df-cau 25426  df-cmet 25427  df-totbnd 38447

This theorem is used by:  heibor  38500