Theorem heibor1 37797

Description: One half of heibor 37808, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 25252 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 766	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐷 ∈ (Met‘𝑋))
3		simplr 768	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝐽 ∈ Comp)
4		simprl 770	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥 ∈ (Cau‘𝐷))
5		simprr 772	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ (Cau‘𝐷) ∧ 𝑥:ℕ⟶𝑋)) → 𝑥:ℕ⟶𝑋)
6	1, 2, 3, 4, 5	heibor1lem 37796	. . . . 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 3125	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑥 ∈ (Cau‘𝐷)(𝑥:ℕ⟶𝑋 → 𝑥 ∈ dom (⇝𝑡‘𝐽)))
9		nnuz 12812	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10		1zzd 12540	. . . 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 25226	. . 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 768	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐽 ∈ Comp)
15		metxmet 24255	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
16		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑋 → 𝑧 ∈ 𝑋)
17		rpxr 12937	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
18	1	blopn 24421	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
19	15, 16, 17, 18	syl3an 1160	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
20	19	3com23 1126	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
21	20	3expa 1118	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → (𝑧(ball‘𝐷)𝑟) ∈ 𝐽)
22		eleq1a 2823	. . . . . . . . . . 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 3134	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
25	24	adantlr 715	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → (∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟) → 𝑦 ∈ 𝐽))
26	25	abssdv 4028	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ 𝐽)
27	15	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑋))
28	1	mopnuni 24362	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	27, 28	syl 17	. . . . . . . . 9 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → 𝑋 = ∪ 𝐽)
30		blcntr 24334	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
31	15, 30	syl3an1 1163	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
32	31	3com23 1126	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+ ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
33	32	3expa 1118	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ (𝑧(ball‘𝐷)𝑟))
34		ovex 7402	. . . . . . . . . . . . . . 15 ⊢ (𝑧(ball‘𝐷)𝑟) ∈ V
35	34	elabrex 7198	. . . . . . . . . . . . . 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 4872	. . . . . . . . . . . . 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 3125	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
40	39	adantlr 715	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∀𝑧 ∈ 𝑋 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
41		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝑋
42		nfre1 3260	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)
43	42	nfab 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧{𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
44	43	nfuni 4874	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)}
45	41, 44	dfss3f 3935	. . . . . . . . . 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 3979	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 ⊆ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
48	26	unissd 4877	. . . . . . . 8 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ⊆ ∪ 𝐽)
49	47, 48	eqssd 3961	. . . . . . 7 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∪ 𝐽 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
50		eqid 2729	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
51	50	cmpcov 23309	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ∧ 𝑟 ∈ ℝ+) → ∃𝑥 ∈ (𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} ∩ Fin)∪ 𝐽 = ∪ 𝑥)
53		elin 3927	. . . . . . . . . 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 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 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 3072	. . . . . 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 2736	. . . . . . . . . 10 ⊢ (∪ 𝑥 = 𝑋 ↔ 𝑋 = ∪ 𝑥)
63	29	eqeq1d 2731	. . . . . . . . . 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 631	. . . . . . . 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 4566	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)} → 𝑥 ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)})
68		ssabral 4025	. . . . . . . . 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 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	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 3148	. . . . 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 3125	. . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → ∀𝑟 ∈ ℝ+ ∃𝑥 ∈ Fin (∪ 𝑥 = 𝑋 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑋 𝑦 = (𝑧(ball‘𝐷)𝑟)))
75		istotbnd 37756	. . 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 511	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867  dom cdm 5631  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Fincfn 8895  1c1 11045  ℝ^*cxr 11183  ℕcn 12162  ℝ⁺crp 12927  ∞Metcxmet 21281  Metcmet 21282  ballcbl 21283  MetOpencmopn 21286  ⇝_𝑡clm 23146  Compccmp 23306  Cauccau 25186  CMetccmet 25187  TotBndctotbnd 37753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cc 10364  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  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 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-omul 8416  df-er 8648  df-map 8778  df-pm 8779  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fi 9338  df-sup 9369  df-inf 9370  df-oi 9439  df-card 9868  df-acn 9871  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-q 12884  df-rp 12928  df-xneg 13048  df-xadd 13049  df-xmul 13050  df-ico 13288  df-fz 13445  df-fl 13730  df-seq 13943  df-exp 14003  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15430  df-rlim 15431  df-rest 17361  df-topgen 17382  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-fbas 21293  df-fg 21294  df-top 22814  df-topon 22831  df-bases 22866  df-cld 22939  df-ntr 22940  df-cls 22941  df-nei 23018  df-lm 23149  df-cmp 23307  df-fil 23766  df-fm 23858  df-flim 23859  df-flf 23860  df-cfil 25188  df-cau 25189  df-cmet 25190  df-totbnd 37755

This theorem is referenced by:  heibor  37808