Theorem heibor 37781

Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 37770 and heiborlem1 37771 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 37770	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3		cmetmet 25339	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	3	adantr 480	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5		metxmet 24365	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	1	mopntop 24471	. . . . . 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 37729	. . . . . . . . . . . . 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 12366	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
12		nnexpcl 14125	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
13	11, 12	mpan 689	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
14	13	nnrpd 13097	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
15	14	rpreccld 13109	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16		oveq2 7456	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
17	16	eqeq2d 2751	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
18	17	rexbidv 3185	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (1 / (2↑𝑛)) → (∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
19	18	ralbidv 3184	. . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . 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 3634	. . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
27	26	eqeq2d 2751	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
28	27	ac6sfi 9348	. . . . . . . . . . . . 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 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 1201	. . . . . . . . . . . . . . . . . . 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 6755	. . . . . . . . . . . . . . . . . 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 24472	. . . . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . 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 4049	. . . . . . . . . . . . . . . . 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 6934	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ∈ V
39	38	uniex 7776	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 ∈ V
40	39	elpw2 5352	. . . . . . . . . . . . . . . . 17 ⊢ (ran 𝑚 ∈ 𝒫 ∪ 𝐽 ↔ ran 𝑚 ⊆ ∪ 𝐽)
41	37, 40	sylibr 234	. . . . . . . . . . . . . . . 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 1199	. . . . . . . . . . . . . . . . 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 6747	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚 Fn 𝑢)
44		dffn4 6840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 Fn 𝑢 ↔ 𝑚:𝑢–onto→ran 𝑚)
45	43, 44	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚:𝑢–onto→ran 𝑚)
46		fofi 9379	. . . . . . . . . . . . . . . . . 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 4223	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
51	50	rexrn 7121	. . . . . . . . . . . . . . . . . . 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 5019	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
53		eliun 5019	. . . . . . . . . . . . . . . . . . 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 314	. . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . . 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 5081	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑢 = ∪ 𝑣 ∈ 𝑢 𝑣
59		iuneq2 5034	. . . . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . . 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 5031	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ran 𝑚 → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
65	64	rspceeqv 3658	. . . . . . . . . . . . . . 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 1121	. . . . . . . . . . . . 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 1932	. . . . . . . . . . 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 3162	. . . . . . . . 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 3160	. . . . . . 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 5398	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝐽 ∈ V
75	74	inex1 5335	. . . . . . . 8 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ∈ V
76		nn0ennn 14030	. . . . . . . . 9 ⊢ ℕ0 ≈ ℕ
77		nnenom 14031	. . . . . . . . 9 ⊢ ℕ ≈ ω
78	76, 77	entri 9068	. . . . . . . 8 ⊢ ℕ0 ≈ ω
79		iuneq1 5031	. . . . . . . . 9 ⊢ (𝑡 = (𝑚‘𝑛) → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
80	79	eqeq2d 2751	. . . . . . . 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 10508	. . . . . . 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 4629	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 𝐽 → 𝑟 ⊆ 𝐽)
84		eqid 2740	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
85		eqid 2740	. . . . . . . . . . . 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 2740	. . . . . . . . . . . 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 4639	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
89	88	ineq1d 4240	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝑋 ∩ Fin) = (𝒫 ∪ 𝐽 ∩ Fin))
90	89	feq3d 6734	. . . . . . . . . . . . . 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 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 7455	. . . . . . . . . . . . . . . . . . 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 5063	. . . . . . . . . . . . . . . . . 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 4258	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 ∪ 𝐽
97	96, 88	sseqtrrid 4062	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋)
98		fss 6763	. . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7118	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ∈ 𝒫 𝑋)
101	100	elpwid 4631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ⊆ 𝑋)
102	101	sselda 4008	. . . . . . . . . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
105		oveq2 7456	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
106	105	oveq2d 7464	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
107	106	oveq2d 7464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
108		ovex 7481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
109	104, 107, 86, 108	ovmpo 7610	. . . . . . . . . . . . . . . . . . . 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 5039	. . . . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . 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 3173	. . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑚‘𝑛) = (𝑚‘𝑘))
118	117	iuneq1d 5042	. . . . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . . 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 5039	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 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 2751	. . . . . . . . . . . . . 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 3243	. . . . . . . . . . . . 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 218	. . . . . . . . . . . 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 37780	. . . . . . . . . . 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 3151	. . . . . . . 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 1932	. . . . . 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 2740	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
135	134	iscmp 23417	. . . 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 209	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931  ∪ ciun 5015  {copab 5228  ran crn 5701   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ωcom 7903  Fincfn 9003  1c1 11185   / cdiv 11947  ℕcn 12293  2c2 12348  ℕ₀cn0 12553  ℝ⁺crp 13057  ↑cexp 14112  ∞Metcxmet 21372  Metcmet 21373  ballcbl 21374  MetOpencmopn 21377  Topctop 22920  Compccmp 23415  CMetccmet 25307  TotBndctotbnd 37726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cc 10504  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-omul 8527  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-acn 10011  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ico 13413  df-icc 13414  df-fz 13568  df-fl 13843  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-rest 17482  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-lm 23258  df-haus 23344  df-cmp 23416  df-fil 23875  df-fm 23967  df-flim 23968  df-flf 23969  df-cfil 25308  df-cau 25309  df-cmet 25310  df-totbnd 37728

This theorem is referenced by:  rrnheibor  37797