Theorem heibor 35259

Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 35248 and heiborlem1 35249 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)

Hypothesis

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

Assertion

Ref	Expression
heibor	⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Proof of Theorem heibor

Dummy variables 𝑡 𝑛 𝑦 𝑘 𝑟 𝑢 𝑚 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		heibor.1	. . 3 ⊢ 𝐽 = (MetOpen‘𝐷)
2	1	heibor1 35248	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3		cmetmet 23890	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	3	adantr 484	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5		metxmet 22941	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	1	mopntop 23047	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7	3, 5, 6	3syl 18	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐽 ∈ Top)
8	7	adantr 484	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Top)
9		istotbnd 35207	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (TotBnd‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑟 ∈ ℝ+ ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟))))
10	9	simprbi 500	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ ℝ+ ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)))
11		2nn 11698	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
12		nnexpcl 13438	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
13	11, 12	mpan 689	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
14	13	nnrpd 12417	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
15	14	rpreccld 12429	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16		oveq2 7143	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
17	16	eqeq2d 2809	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
18	17	rexbidv 3256	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (1 / (2↑𝑛)) → (∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
19	18	ralbidv 3162	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (1 / (2↑𝑛)) → (∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
20	19	anbi2d 631	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (1 / (2↑𝑛)) → ((∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)) ↔ (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
21	20	rexbidv 3256	. . . . . . . . . . . . 13 ⊢ (𝑟 = (1 / (2↑𝑛)) → (∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)) ↔ ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
22	21	rspccva 3570	. . . . . . . . . . . 12 ⊢ ((∀𝑟 ∈ ℝ+ ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)) ∧ (1 / (2↑𝑛)) ∈ ℝ+) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
23	10, 15, 22	syl2an 598	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (TotBnd‘𝑋) ∧ 𝑛 ∈ ℕ0) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
24	23	expcom 417	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
25	24	adantl 485	. . . . . . . . 9 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
26		oveq1 7142	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
27	26	eqeq2d 2809	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
28	27	ac6sfi 8746	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Fin ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑚(𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
29	28	adantrl 715	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Fin ∧ (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → ∃𝑚(𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
30	29	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))) → ∃𝑚(𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
31		simp3l 1198	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑚:𝑢⟶𝑋)
32	31	frnd 6494	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ⊆ 𝑋)
33	1	mopnuni 23048	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
34	3, 5, 33	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 = ∪ 𝐽)
35	34	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋 = ∪ 𝐽)
36	35	3ad2ant1 1130	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑋 = ∪ 𝐽)
37	32, 36	sseqtrd 3955	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ⊆ ∪ 𝐽)
38	1	fvexi 6659	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ∈ V
39	38	uniex 7447	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 ∈ V
40	39	elpw2 5212	. . . . . . . . . . . . . . . . 17 ⊢ (ran 𝑚 ∈ 𝒫 ∪ 𝐽 ↔ ran 𝑚 ⊆ ∪ 𝐽)
41	37, 40	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ∈ 𝒫 ∪ 𝐽)
42		simp2l 1196	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑢 ∈ Fin)
43		ffn 6487	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚 Fn 𝑢)
44		dffn4 6571	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 Fn 𝑢 ↔ 𝑚:𝑢–onto→ran 𝑚)
45	43, 44	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚:𝑢–onto→ran 𝑚)
46		fofi 8794	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ Fin ∧ 𝑚:𝑢–onto→ran 𝑚) → ran 𝑚 ∈ Fin)
47	45, 46	sylan2 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ Fin ∧ 𝑚:𝑢⟶𝑋) → ran 𝑚 ∈ Fin)
48	42, 31, 47	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ∈ Fin)
49	41, 48	elind 4121	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ran 𝑚 ∈ (𝒫 ∪ 𝐽 ∩ Fin))
50	26	eleq2d 2875	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
51	50	rexrn 6830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 Fn 𝑢 → (∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣 ∈ 𝑢 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
52		eliun 4885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
53		eliun 4885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣 ∈ 𝑢 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
54	51, 52, 53	3bitr4g 317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 Fn 𝑢 → (𝑟 ∈ ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
55	54	eqrdv 2796	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 Fn 𝑢 → ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
56	31, 43, 55	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
57		simp3r 1199	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
58		uniiun 4945	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑢 = ∪ 𝑣 ∈ 𝑢 𝑣
59		iuneq2 4900	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → ∪ 𝑣 ∈ 𝑢 𝑣 = ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
60	58, 59	syl5eq 2845	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → ∪ 𝑢 = ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
61	57, 60	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ∪ 𝑢 = ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
62		simp2r 1197	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ∪ 𝑢 = 𝑋)
63	56, 61, 62	3eqtr2rd 2840	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑋 = ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
64		iuneq1 4897	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ran 𝑚 → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
65	64	rspceeqv 3586	. . . . . . . . . . . . . . 15 ⊢ ((ran 𝑚 ∈ (𝒫 ∪ 𝐽 ∩ Fin) ∧ 𝑋 = ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
66	49, 63, 65	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋) ∧ (𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
67	66	3expia 1118	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ ∪ 𝑢 = 𝑋)) → ((𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
68	67	adantrrr 724	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))) → ((𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
69	68	exlimdv 1934	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))) → (∃𝑚(𝑚:𝑢⟶𝑋 ∧ ∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
70	30, 69	mpd 15	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) ∧ (𝑢 ∈ Fin ∧ (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
71	70	rexlimdvaa 3244	. . . . . . . . 9 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → (∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
72	25, 71	syld 47	. . . . . . . 8 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
73	72	ralrimdva 3154	. . . . . . 7 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑛 ∈ ℕ0 ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
74	39	pwex 5246	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝐽 ∈ V
75	74	inex1 5185	. . . . . . . 8 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ∈ V
76		nn0ennn 13342	. . . . . . . . 9 ⊢ ℕ0 ≈ ℕ
77		nnenom 13343	. . . . . . . . 9 ⊢ ℕ ≈ ω
78	76, 77	entri 8546	. . . . . . . 8 ⊢ ℕ0 ≈ ω
79		iuneq1 4897	. . . . . . . . 9 ⊢ (𝑡 = (𝑚‘𝑛) → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
80	79	eqeq2d 2809	. . . . . . . 8 ⊢ (𝑡 = (𝑚‘𝑛) → (𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
81	75, 78, 80	axcc4 9850	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ0 ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) → ∃𝑚(𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
82	73, 81	syl6 35	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑚(𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
83		elpwi 4506	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 𝐽 → 𝑟 ⊆ 𝐽)
84		eqid 2798	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
85		eqid 2798	. . . . . . . . . . . 12 ⊢ {⟨𝑡, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ 𝑡 ∈ (𝑚‘𝑘) ∧ (𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘) ∈ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣})} = {⟨𝑡, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ 𝑡 ∈ (𝑚‘𝑘) ∧ (𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘) ∈ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣})}
86		eqid 2798	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
87		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → 𝐷 ∈ (CMet‘𝑋))
88	34	pweqd 4516	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
89	88	ineq1d 4138	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝑋 ∩ Fin) = (𝒫 ∪ 𝐽 ∩ Fin))
90	89	feq3d 6474	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin) ↔ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)))
91	90	biimpar 481	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin))
92	91	adantrr 716	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → 𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin))
93		oveq1 7142	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑦 → (𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛))
94	93	cbviunv 4927	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛)
95		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) → 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin))
96		inss1 4155	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 ∪ 𝐽
97	96, 88	sseqtrrid 3968	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋)
98		fss 6501	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋) → 𝑚:ℕ0⟶𝒫 𝑋)
99	95, 97, 98	syl2anr 599	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶𝒫 𝑋)
100	99	ffvelrnda 6828	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ∈ 𝒫 𝑋)
101	100	elpwid 4508	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ⊆ 𝑋)
102	101	sselda 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → 𝑦 ∈ 𝑋)
103		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → 𝑛 ∈ ℕ0)
104		oveq1 7142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
105		oveq2 7143	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
106	105	oveq2d 7151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
107	106	oveq2d 7151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
108		ovex 7168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
109	104, 107, 86, 108	ovmpo 7289	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑛 ∈ ℕ0) → (𝑦(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
110	102, 103, 109	syl2anc 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → (𝑦(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
111	110	iuneq2dv 4905	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
112	94, 111	syl5eq 2845	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
113	112	eqeq2d 2809	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) ↔ 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
114	113	biimprd 251	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))) → 𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛)))
115	114	ralimdva 3144	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → (∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))) → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛)))
116	115	impr 458	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛))
117		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑚‘𝑛) = (𝑚‘𝑘))
118	117	iuneq1d 4908	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = ∪ 𝑡 ∈ (𝑚‘𝑘)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛))
119		simpl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 = 𝑘 ∧ 𝑡 ∈ (𝑚‘𝑘)) → 𝑛 = 𝑘)
120	119	oveq2d 7151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 = 𝑘 ∧ 𝑡 ∈ (𝑚‘𝑘)) → (𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
121	120	iuneq2dv 4905	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → ∪ 𝑡 ∈ (𝑚‘𝑘)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = ∪ 𝑡 ∈ (𝑚‘𝑘)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
122	118, 121	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = ∪ 𝑡 ∈ (𝑚‘𝑘)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
123	122	eqeq2d 2809	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) ↔ 𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑘)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘)))
124	123	cbvralvw 3396	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑛)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) ↔ ∀𝑘 ∈ ℕ0 𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑘)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
125	116, 124	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → ∀𝑘 ∈ ℕ0 𝑋 = ∪ 𝑡 ∈ (𝑚‘𝑘)(𝑡(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑘))
126	1, 84, 85, 86, 87, 92, 125	heiborlem10 35258	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) ∧ (𝑟 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑟)) → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)
127	126	exp32 424	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → (𝑟 ⊆ 𝐽 → (∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
128	83, 127	syl5 34	. . . . . . . . 9 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → (𝑟 ∈ 𝒫 𝐽 → (∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
129	128	ralrimiv 3148	. . . . . . . 8 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣))
130	129	ex 416	. . . . . . 7 ⊢ (𝐷 ∈ (CMet‘𝑋) → ((𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
131	130	exlimdv 1934	. . . . . 6 ⊢ (𝐷 ∈ (CMet‘𝑋) → (∃𝑚(𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
132	82, 131	syld 47	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
133	132	imp 410	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣))
134		eqid 2798	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
135	134	iscmp 21993	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
136	8, 133, 135	sylanbrc 586	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Comp)
137	4, 136	jca 515	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp))
138	2, 137	impbii 212	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  ∪ ciun 4881  {copab 5092  ran crn 5520   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  ωcom 7560  Fincfn 8492  1c1 10527   / cdiv 11286  ℕcn 11625  2c2 11680  ℕ₀cn0 11885  ℝ⁺crp 12377  ↑cexp 13425  ∞Metcxmet 20076  Metcmet 20077  ballcbl 20078  MetOpencmopn 20081  Topctop 21498  Compccmp 21991  CMetccmet 23858  TotBndctotbnd 35204

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cc 9846  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-acn 9355  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ico 12732  df-icc 12733  df-fz 12886  df-fl 13157  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-rest 16688  df-topgen 16709  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-fbas 20088  df-fg 20089  df-top 21499  df-topon 21516  df-bases 21551  df-cld 21624  df-ntr 21625  df-cls 21626  df-nei 21703  df-lm 21834  df-haus 21920  df-cmp 21992  df-fil 22451  df-fm 22543  df-flim 22544  df-flf 22545  df-cfil 23859  df-cau 23860  df-cmet 23861  df-totbnd 35206

This theorem is referenced by:  rrnheibor  35275