Theorem heibor 35906

Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 35895 and heiborlem1 35896 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 35895	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3		cmetmet 24355	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	3	adantr 480	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5		metxmet 23395	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	1	mopntop 23501	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7	3, 5, 6	3syl 18	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐽 ∈ Top)
8	7	adantr 480	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Top)
9		istotbnd 35854	. . . . . . . . . . . . 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 496	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ ℝ+ ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)))
11		2nn 11976	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
12		nnexpcl 13723	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
13	11, 12	mpan 686	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
14	13	nnrpd 12699	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
15	14	rpreccld 12711	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16		oveq2 7263	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
17	16	eqeq2d 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
18	17	rexbidv 3225	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (1 / (2↑𝑛)) → (∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
19	18	ralbidv 3120	. . . . . . . . . . . . . . 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 628	. . . . . . . . . . . . . 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 3225	. . . . . . . . . . . . 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 3551	. . . . . . . . . . . 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 595	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (TotBnd‘𝑋) ∧ 𝑛 ∈ ℕ0) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
24	23	expcom 413	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
25	24	adantl 481	. . . . . . . . 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 7262	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
27	26	eqeq2d 2749	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
28	27	ac6sfi 8988	. . . . . . . . . . . . 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 712	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 1199	. . . . . . . . . . . . . . . . . . 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 6592	. . . . . . . . . . . . . . . . . 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 23502	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
34	3, 5, 33	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 = ∪ 𝐽)
35	34	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋 = ∪ 𝐽)
36	35	3ad2ant1 1131	. . . . . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . . . . . 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 6770	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ∈ V
39	38	uniex 7572	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 ∈ V
40	39	elpw2 5264	. . . . . . . . . . . . . . . . 17 ⊢ (ran 𝑚 ∈ 𝒫 ∪ 𝐽 ↔ ran 𝑚 ⊆ ∪ 𝐽)
41	37, 40	sylibr 233	. . . . . . . . . . . . . . . 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 1197	. . . . . . . . . . . . . . . . 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 6584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚 Fn 𝑢)
44		dffn4 6678	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 Fn 𝑢 ↔ 𝑚:𝑢–onto→ran 𝑚)
45	43, 44	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚:𝑢–onto→ran 𝑚)
46		fofi 9035	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ Fin ∧ 𝑚:𝑢–onto→ran 𝑚) → ran 𝑚 ∈ Fin)
47	45, 46	sylan2 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ Fin ∧ 𝑚:𝑢⟶𝑋) → ran 𝑚 ∈ Fin)
48	42, 31, 47	syl2anc 583	. . . . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
51	50	rexrn 6945	. . . . . . . . . . . . . . . . . . 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 4925	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
53		eliun 4925	. . . . . . . . . . . . . . . . . . 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 313	. . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . . . . 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 4984	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑢 = ∪ 𝑣 ∈ 𝑢 𝑣
59		iuneq2 4940	. . . . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . . . 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 1198	. . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . 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 4937	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ran 𝑚 → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
65	64	rspceeqv 3567	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . 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 721	. . . . . . . . . . . 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 1937	. . . . . . . . . . 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 3213	. . . . . . . . 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 3112	. . . . . . 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 5298	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝐽 ∈ V
75	74	inex1 5236	. . . . . . . 8 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ∈ V
76		nn0ennn 13627	. . . . . . . . 9 ⊢ ℕ0 ≈ ℕ
77		nnenom 13628	. . . . . . . . 9 ⊢ ℕ ≈ ω
78	76, 77	entri 8749	. . . . . . . 8 ⊢ ℕ0 ≈ ω
79		iuneq1 4937	. . . . . . . . 9 ⊢ (𝑡 = (𝑚‘𝑛) → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
80	79	eqeq2d 2749	. . . . . . . 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 10126	. . . . . . 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 4539	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 𝐽 → 𝑟 ⊆ 𝐽)
84		eqid 2738	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
85		eqid 2738	. . . . . . . . . . . 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 2738	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
87		simpl 482	. . . . . . . . . . . 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 4549	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
89	88	ineq1d 4142	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝑋 ∩ Fin) = (𝒫 ∪ 𝐽 ∩ Fin))
90	89	feq3d 6571	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin) ↔ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)))
91	90	biimpar 477	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin))
92	91	adantrr 713	. . . . . . . . . . . 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 7262	. . . . . . . . . . . . . . . . . . 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 4966	. . . . . . . . . . . . . . . . . 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 4159	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 ∪ 𝐽
97	96, 88	sseqtrrid 3970	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋)
98		fss 6601	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋) → 𝑚:ℕ0⟶𝒫 𝑋)
99	95, 97, 98	syl2anr 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶𝒫 𝑋)
100	99	ffvelrnda 6943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ∈ 𝒫 𝑋)
101	100	elpwid 4541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ⊆ 𝑋)
102	101	sselda 3917	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → 𝑦 ∈ 𝑋)
103		simplr 765	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → 𝑛 ∈ ℕ0)
104		oveq1 7262	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
105		oveq2 7263	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
106	105	oveq2d 7271	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
107	106	oveq2d 7271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
108		ovex 7288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
109	104, 107, 86, 108	ovmpo 7411	. . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . 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 4945	. . . . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . 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 3102	. . . . . . . . . . . . . 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 454	. . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑚‘𝑛) = (𝑚‘𝑘))
118	117	iuneq1d 4948	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 = 𝑘 ∧ 𝑡 ∈ (𝑚‘𝑘)) → 𝑛 = 𝑘)
120	119	oveq2d 7271	. . . . . . . . . . . . . . . . 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 4945	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . . 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 3372	. . . . . . . . . . . . 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 217	. . . . . . . . . . . 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 35905	. . . . . . . . . . 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 420	. . . . . . . . . 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 3106	. . . . . . . 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 412	. . . . . . 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 1937	. . . . . 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 406	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣))
134		eqid 2738	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
135	134	iscmp 22447	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
136	8, 133, 135	sylanbrc 582	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Comp)
137	4, 136	jca 511	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp))
138	2, 137	impbii 208	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ∪ ciun 4921  {copab 5132  ran crn 5581   Fn wfn 6413  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  ωcom 7687  Fincfn 8691  1c1 10803   / cdiv 11562  ℕcn 11903  2c2 11958  ℕ₀cn0 12163  ℝ⁺crp 12659  ↑cexp 13710  ∞Metcxmet 20495  Metcmet 20496  ballcbl 20497  MetOpencmopn 20500  Topctop 21950  Compccmp 22445  CMetccmet 24323  TotBndctotbnd 35851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cc 10122  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-acn 9631  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ico 13014  df-icc 13015  df-fz 13169  df-fl 13440  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-rest 17050  df-topgen 17071  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-fbas 20507  df-fg 20508  df-top 21951  df-topon 21968  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-lm 22288  df-haus 22374  df-cmp 22446  df-fil 22905  df-fm 22997  df-flim 22998  df-flf 22999  df-cfil 24324  df-cau 24325  df-cmet 24326  df-totbnd 35853

This theorem is referenced by:  rrnheibor  35922