Theorem heibor1 32779

Description: One half of heibor 32790, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 22924 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 786	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐷 ∈ (Met‘𝑋))
3		simplr 788	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐽 ∈ Comp)
4		simprl 790	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ (Cau‘𝐷))
5		simprr 792	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥:ℕ⟶𝑋)
6	1, 2, 3, 4, 5	heibor1lem 32778	. . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ dom (⇝𝑡‘𝐽))
7	6	expr 641	. . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑥 ∈ (Cau‘𝐷)) → (𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
8	7	ralrimiva 2949	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
9		nnuz 11599	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10		1zzd 11285	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 1 ∈ ℤ)
11		simpl 472	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (Met‘𝑋))
12	9, 1, 10, 11	iscmet3 22899	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽))))
13	8, 12	mpbird 246	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (CMet‘𝑋))
14		simplr 788	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐽 ∈ Comp)
15		metxmet 21949	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
16		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → 𝑧 ∈ 𝑋)
17		rpxr 11716	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
18	1	blopn 22115	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
19	15, 16, 17, 18	syl3an 1360	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
20	19	3com23 1263	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
21	20	3expa 1257	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
22		eleq1a 2683	. . . . . . . . . . 11 ⊢ ((𝑧(ball‘𝐷)𝑟) ∈ 𝐽 → (𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
23	21, 22	syl 17	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
24	23	rexlimdva 3013	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
25	24	adantlr 747	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
26	25	abssdv 3639	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ 𝐽)
27	15	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑋))
28	1	mopnuni 22056	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	27, 28	syl 17	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 = ∪ 𝐽)
30		blcntr 22028	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
31	15, 30	syl3an1 1351	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
32	31	3com23 1263	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
33	32	3expa 1257	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
34		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (𝑧(ball‘𝐷)𝑟) ∈ V
35	34	elabrex 6405	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → (𝑧(ball‘𝐷)𝑟) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
36	35	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
37		elunii 4377	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
39	38	ralrimiva 2949	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
40	39	adantlr 747	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
41		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝑋
42		nfre1 2988	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)
43	42	nfab 2755	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧{𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
44	43	nfuni 4378	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
45	41, 44	dfss3f 3560	. . . . . . . . . 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 223	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
47	29, 46	eqsstr3d 3603	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
48	26	unissd 4398	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ ∪ 𝐽)
49	47, 48	eqssd 3585	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
50		eqid 2610	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
51	50	cmpcov 21002	. . . . . . 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 1318	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin)∪ 𝐽 = ∪ 𝑥)
53		elin 3758	. . . . . . . . . 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 263	. . . . . . . . 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 727	. . . . . . . 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 679	. . . . . . . 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 263	. . . . . . 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 3021	. . . . . 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 207	. . . . 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 2617	. . . . . . . . . 10 ⊢ (∪ 𝑥 = 𝑋 ↔ 𝑋 = ∪ 𝑥)
63	29	eqeq1d 2612	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (𝑋 = ∪ 𝑥 ↔ ∪ 𝐽 = ∪ 𝑥))
64	62, 63	syl5rbb 272	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∪ 𝐽 = ∪ 𝑥 ↔ ∪ 𝑥 = 𝑋))
65	64	anbi1d 737	. . . . . . . 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	syl5bb 271	. . . . . . 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 4117	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → 𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
68		ssabral 3636	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))
69	67, 68	sylib 207	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))
70	69	anim2i 591	. . . . . . 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	syl6bi 242	. . . . . 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 2999	. . . . 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 15	. . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)))
74	73	ralrimiva 2949	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑟 ∈ ℝ+ ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)))
75		istotbnd 32738	. . 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 695	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (TotBnd‘𝑋))
77	13, 76	jca 553	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816  ℝ^*cxr 9952  ℕcn 10897  ℝ⁺crp 11708  ∞Metcxmt 19552  Metcme 19553  ballcbl 19554  MetOpencmopn 19557  ⇝_𝑡clm 20840  Compccmp 20999  Caucca 22859  CMetcms 22860  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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  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  ax-pre-sup 9893

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-iin 4458  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-se 4998  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-isom 5813  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-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  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-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-fz 12198  df-fl 12455  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lm 20843  df-cmp 21000  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-cfil 22861  df-cau 22862  df-cmet 22863  df-totbnd 32737

This theorem is referenced by:  heibor  32790