Theorem heibor 38755

Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 38744 and heiborlem1 38745 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 38744	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3		cmetmet 25607	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	3	adantr 486	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5		metxmet 24653	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	1	mopntop 24759	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7	3, 5, 6	3syl 19	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐽 ∈ Top)
8	7	adantr 486	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Top)
9		istotbnd 38703	. . . . . . . . . . . . 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 503	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ ℝ+ ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)))
11		2nn 12416	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
12		nnexpcl 14217	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
13	11, 12	mpan 703	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
14	13	nnrpd 13162	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
15	14	rpreccld 13174	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16		oveq2 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
17	16	eqeq2d 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
18	17	rexbidv 3187	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (1 / (2↑𝑛)) → (∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
19	18	ralbidv 3186	. . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . 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 3187	. . . . . . . . . . . . 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 3576	. . . . . . . . . . . 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 608	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (TotBnd‘𝑋) ∧ 𝑛 ∈ ℕ0) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
24	23	expcom 419	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
25	24	adantl 487	. . . . . . . . 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 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
27	26	eqeq2d 2772	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
28	27	ac6sfi 9275	. . . . . . . . . . . . 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 729	. . . . . . . . . . . 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 487	. . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . . . . 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 6718	. . . . . . . . . . . . . . . . . 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 24760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
34	3, 5, 33	3syl 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 = ∪ 𝐽)
35	34	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋 = ∪ 𝐽)
36	35	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . . . 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 6899	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ∈ V
39	38	uniex 7758	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 ∈ V
40	39	elpw2 5296	. . . . . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . . . . 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 6709	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚 Fn 𝑢)
44		dffn4 6802	. . . . . . . . . . . . . . . . . . 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 9305	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ Fin ∧ 𝑚:𝑢–onto→ran 𝑚) → ran 𝑚 ∈ Fin)
47	45, 46	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ Fin ∧ 𝑚:𝑢⟶𝑋) → ran 𝑚 ∈ Fin)
48	42, 31, 47	syl2anc 596	. . . . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
51	50	rexrn 7087	. . . . . . . . . . . . . . . . . . 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 4955	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
53		eliun 4955	. . . . . . . . . . . . . . . . . . 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 2759	. . . . . . . . . . . . . . . . 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 19	. . . . . . . . . . . . . . . 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 1221	. . . . . . . . . . . . . . . . 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 5017	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑢 = ∪ 𝑣 ∈ 𝑢 𝑣
59		iuneq2 4971	. . . . . . . . . . . . . . . . . 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	eqtrid 2808	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 1219	. . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . 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 4968	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ran 𝑚 → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
65	64	rspceeqv 3599	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . 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 738	. . . . . . . . . . . 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 1966	. . . . . . . . . . 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 16	. . . . . . . . . 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 3165	. . . . . . . . 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 48	. . . . . . . 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 3163	. . . . . . 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 5342	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝐽 ∈ V
75	74	inex1 5277	. . . . . . . 8 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ∈ V
76		nn0ennn 14122	. . . . . . . . 9 ⊢ ℕ0 ≈ ℕ
77		nnenom 14123	. . . . . . . . 9 ⊢ ℕ ≈ ω
78	76, 77	entri 9035	. . . . . . . 8 ⊢ ℕ0 ≈ ω
79		iuneq1 4968	. . . . . . . . 9 ⊢ (𝑡 = (𝑚‘𝑛) → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
80	79	eqeq2d 2772	. . . . . . . 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 10517	. . . . . . 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 36	. . . . . 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 4564	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 𝐽 → 𝑟 ⊆ 𝐽)
84		eqid 2761	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
85		eqid 2761	. . . . . . . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
87		simpl 488	. . . . . . . . . . . 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 4574	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
89	88	ineq1d 4165	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝑋 ∩ Fin) = (𝒫 ∪ 𝐽 ∩ Fin))
90	89	feq3d 6694	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin) ↔ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)))
91	90	biimpar 483	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin))
92	91	adantrr 730	. . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . 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 4997	. . . . . . . . . . . . . . . . . 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 23	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) → 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin))
96		inss1 4182	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 ∪ 𝐽
97	96, 88	sseqtrrid 3974	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋)
98		fss 6726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋) → 𝑚:ℕ0⟶𝒫 𝑋)
99	95, 97, 98	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶𝒫 𝑋)
100	99	ffvelcdmda 7084	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ∈ 𝒫 𝑋)
101	100	elpwid 4566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ⊆ 𝑋)
102	101	sselda 3931	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → 𝑦 ∈ 𝑋)
103		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → 𝑛 ∈ ℕ0)
104		oveq1 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
105		oveq2 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
106	105	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
107	106	oveq2d 7436	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
108		ovex 7453	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
109	104, 107, 86, 108	ovmpo 7580	. . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . 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 4976	. . . . . . . . . . . . . . . . . 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	eqtrid 2808	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . . 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 460	. . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑚‘𝑛) = (𝑚‘𝑘))
118	117	iuneq1d 4979	. . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 = 𝑘 ∧ 𝑡 ∈ (𝑚‘𝑘)) → 𝑛 = 𝑘)
120	119	oveq2d 7436	. . . . . . . . . . . . . . . . 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 4976	. . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 3241	. . . . . . . . . . . . 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 38754	. . . . . . . . . . 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 426	. . . . . . . . . 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 35	. . . . . . . . 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 3154	. . . . . . . 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 418	. . . . . . 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 1966	. . . . . 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 48	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
133	132	imp 412	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣))
134		eqid 2761	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
135	134	iscmp 23706	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
136	8, 133, 135	sylanbrc 595	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Comp)
137	4, 136	jca 521	. 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ∪ ciun 4951  {copab 5167  ran crn 5652   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  ωcom 7877  Fincfn 8973  1c1 11201   / cdiv 11973  ℕcn 12335  2c2 12397  ℕ₀cn0 12606  ℝ⁺crp 13120  ↑cexp 14204  ∞Metcxmet 21663  Metcmet 21664  ballcbl 21665  MetOpencmopn 21668  Topctop 23211  Compccmp 23704  CMetccmet 25575  TotBndctotbnd 38700

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-cc 10513  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-omul 8481  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9504  df-card 10020  df-acn 10023  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-ico 13482  df-icc 13483  df-fz 13640  df-fl 13932  df-seq 14145  df-exp 14205  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-clim 15655  df-rlim 15656  df-rest 17593  df-topgen 17614  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-fbas 21675  df-fg 21676  df-top 23212  df-topon 23229  df-bases 23264  df-cld 23337  df-ntr 23338  df-cls 23339  df-nei 23416  df-lm 23547  df-haus 23633  df-cmp 23705  df-fil 24165  df-fm 24257  df-flim 24258  df-flf 24259  df-cfil 25576  df-cau 25577  df-cmet 25578  df-totbnd 38702

This theorem is used by:  rrnheibor  38771