Theorem heibor1 35968

Description: One half of heibor 35979, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 24483 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 764	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐷 ∈ (Met‘𝑋))
3		simplr 766	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐽 ∈ Comp)
4		simprl 768	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ (Cau‘𝐷))
5		simprr 770	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥:ℕ⟶𝑋)
6	1, 2, 3, 4, 5	heibor1lem 35967	. . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ dom (⇝𝑡‘𝐽))
7	6	expr 457	. . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑥 ∈ (Cau‘𝐷)) → (𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
8	7	ralrimiva 3103	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
9		nnuz 12621	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10		1zzd 12351	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 1 ∈ ℤ)
11		simpl 483	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (Met‘𝑋))
12	9, 1, 10, 11	iscmet3 24457	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽))))
13	8, 12	mpbird 256	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (CMet‘𝑋))
14		simplr 766	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐽 ∈ Comp)
15		metxmet 23487	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
16		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → 𝑧 ∈ 𝑋)
17		rpxr 12739	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
18	1	blopn 23656	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
19	15, 16, 17, 18	syl3an 1159	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
20	19	3com23 1125	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
21	20	3expa 1117	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
22		eleq1a 2834	. . . . . . . . . . 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 3213	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
25	24	adantlr 712	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
26	25	abssdv 4002	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ 𝐽)
27	15	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑋))
28	1	mopnuni 23594	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	27, 28	syl 17	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 = ∪ 𝐽)
30		blcntr 23566	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
31	15, 30	syl3an1 1162	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
32	31	3com23 1125	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
33	32	3expa 1117	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
34		ovex 7308	. . . . . . . . . . . . . . 15 ⊢ (𝑧(ball‘𝐷)𝑟) ∈ V
35	34	elabrex 7116	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → (𝑧(ball‘𝐷)𝑟) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
36	35	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
37		elunii 4844	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
39	38	ralrimiva 3103	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
40	39	adantlr 712	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
41		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝑋
42		nfre1 3239	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)
43	42	nfab 2913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧{𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
44	43	nfuni 4846	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
45	41, 44	dfss3f 3912	. . . . . . . . . 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 233	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
47	29, 46	eqsstrrd 3960	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
48	26	unissd 4849	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ ∪ 𝐽)
49	47, 48	eqssd 3938	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
50		eqid 2738	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
51	50	cmpcov 22540	. . . . . . 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 1370	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin)∪ 𝐽 = ∪ 𝑥)
53		elin 3903	. . . . . . . . . 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 461	. . . . . . . . . 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 274	. . . . . . . . 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 624	. . . . . . . 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 469	. . . . . . . 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 274	. . . . . . 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 3179	. . . . . 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 217	. . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ Fin (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∧ ∪ 𝐽 = ∪ 𝑥))
61		ancom 461	. . . . . . . 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 2745	. . . . . . . . . 10 ⊢ (∪ 𝑥 = 𝑋 ↔ 𝑋 = ∪ 𝑥)
63	29	eqeq1d 2740	. . . . . . . . . 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 630	. . . . . . . 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 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 4542	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → 𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
68		ssabral 3996	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))
69	67, 68	sylib 217	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟))
70	69	anim2i 617	. . . . . . 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 252	. . . . . 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 3202	. . . . 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 3103	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑟 ∈ ℝ+ ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)))
75		istotbnd 35927	. . 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 583	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → 𝐷 ∈ (TotBnd‘𝑋))
77	13, 76	jca 512	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  dom cdm 5589  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  1c1 10872  ℝ^*cxr 11008  ℕcn 11973  ℝ⁺crp 12730  ∞Metcxmet 20582  Metcmet 20583  ballcbl 20584  MetOpencmopn 20587  ⇝_𝑡clm 22377  Compccmp 22537  Cauccau 24417  CMetccmet 24418  TotBndctotbnd 35924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-acn 9700  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ico 13085  df-fz 13240  df-fl 13512  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lm 22380  df-cmp 22538  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-cfil 24419  df-cau 24420  df-cmet 24421  df-totbnd 35926

This theorem is referenced by:  heibor  35979