Theorem heibor1 38131

Description: One half of heibor 38142, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 25286 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 767	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐷 ∈ (Met‘𝑋))
3		simplr 769	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐽 ∈ Comp)
4		simprl 771	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ (Cau‘𝐷))
5		simprr 773	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥:ℕ⟶𝑋)
6	1, 2, 3, 4, 5	heibor1lem 38130	. . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ dom (⇝𝑡‘𝐽))
7	6	expr 456	. . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑥 ∈ (Cau‘𝐷)) → (𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
8	7	ralrimiva 3129	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
9		nnuz 12827	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10		1zzd 12558	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 1 ∈ ℤ)
11		simpl 482	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (Met‘𝑋))
12	9, 1, 10, 11	iscmet3 25260	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽))))
13	8, 12	mpbird 257	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (CMet‘𝑋))
14		simplr 769	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐽 ∈ Comp)
15		metxmet 24299	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
16		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → 𝑧 ∈ 𝑋)
17		rpxr 12952	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
18	1	blopn 24465	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
19	15, 16, 17, 18	syl3an 1161	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
20	19	3com23 1127	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
21	20	3expa 1119	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
22		eleq1a 2831	. . . . . . . . . . 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 3138	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
25	24	adantlr 716	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
26	25	abssdv 4007	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ 𝐽)
27	15	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑋))
28	1	mopnuni 24406	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	27, 28	syl 17	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 = ∪ 𝐽)
30		blcntr 24378	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
31	15, 30	syl3an1 1164	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
32	31	3com23 1127	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
33	32	3expa 1119	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
34		ovex 7400	. . . . . . . . . . . . . . 15 ⊢ (𝑧(ball‘𝐷)𝑟) ∈ V
35	34	elabrex 7197	. . . . . . . . . . . . . 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 4855	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
39	38	ralrimiva 3129	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
40	39	adantlr 716	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
41		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝑋
42		nfre1 3262	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)
43	42	nfab 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧{𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
44	43	nfuni 4857	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
45	41, 44	dfss3f 3913	. . . . . . . . . 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 234	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
47	29, 46	eqsstrrd 3957	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
48	26	unissd 4860	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ ∪ 𝐽)
49	47, 48	eqssd 3939	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
50		eqid 2736	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
51	50	cmpcov 23354	. . . . . . 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 1374	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin)∪ 𝐽 = ∪ 𝑥)
53		elin 3905	. . . . . . . . . 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 460	. . . . . . . . . 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 275	. . . . . . . . 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 625	. . . . . . . 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 468	. . . . . . . 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 275	. . . . . . 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 3080	. . . . . 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 218	. . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ Fin (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥))
61		ancom 460	. . . . . . . 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 2743	. . . . . . . . . 10 ⊢ (∪ 𝑥 = 𝑋 ↔ 𝑋 = ∪ 𝑥)
63	29	eqeq1d 2738	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (𝑋 = ∪ 𝑥 ↔ ∪ 𝐽 = ∪ 𝑥))
64	62, 63	bitr2id 284	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∪ 𝐽 = ∪ 𝑥 ↔ ∪ 𝑥 = 𝑋))
65	64	anbi1d 632	. . . . . . . 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 283	. . . . . . 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 4548	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → 𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
68		ssabral 4004	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))
69	67, 68	sylib 218	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))
70	69	anim2i 618	. . . . . . 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 253	. . . . . 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 3152	. . . . 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 3129	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑟 ∈ ℝ+ ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)))
75		istotbnd 38090	. . 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 584	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (TotBnd‘𝑋))
77	13, 76	jca 511	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  dom cdm 5631  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Fincfn 8893  1c1 11039  ℝ^*cxr 11178  ℕcn 12174  ℝ⁺crp 12942  ∞Metcxmet 21337  Metcmet 21338  ballcbl 21339  MetOpencmopn 21342  ⇝_𝑡clm 23191  Compccmp 23351  Cauccau 25220  CMetccmet 25221  TotBndctotbnd 38087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cc 10357  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-omul 8410  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fi 9324  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-acn 9866  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-ico 13304  df-fz 13462  df-fl 13751  df-seq 13964  df-exp 14024  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-rlim 15451  df-rest 17385  df-topgen 17406  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-fbas 21349  df-fg 21350  df-top 22859  df-topon 22876  df-bases 22911  df-cld 22984  df-ntr 22985  df-cls 22986  df-nei 23063  df-lm 23194  df-cmp 23352  df-fil 23811  df-fm 23903  df-flim 23904  df-flf 23905  df-cfil 25222  df-cau 25223  df-cmet 25224  df-totbnd 38089

This theorem is referenced by:  heibor  38142