Theorem heibor 33620

Description: Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 33609 and heiborlem1 33610 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 33609	. 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
3		cmetmet 23084	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	3	adantr 481	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐷 ∈ (Met‘𝑋))
5		metxmet 22139	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	1	mopntop 22245	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
7	3, 5, 6	3syl 18	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐽 ∈ Top)
8	7	adantr 481	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Top)
9		istotbnd 33568	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ ℝ+ ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟)))
11		2nn 11185	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
12		nnexpcl 12873	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
13	11, 12	mpan 706	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℕ)
14	13	nnrpd 11870	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 → (2↑𝑛) ∈ ℝ+)
15	14	rpreccld 11882	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 → (1 / (2↑𝑛)) ∈ ℝ+)
16		oveq2 6658	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
17	16	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (1 / (2↑𝑛)) → (𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
18	17	rexbidv 3052	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (1 / (2↑𝑛)) → (∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
19	18	ralbidv 2986	. . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . 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 3052	. . . . . . . . . . . . 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 3308	. . . . . . . . . . . 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 494	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (TotBnd‘𝑋) ∧ 𝑛 ∈ ℕ0) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
24	23	expcom 451	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑢 ∈ Fin (∪ 𝑢 = 𝑋 ∧ ∀𝑣 ∈ 𝑢 ∃𝑦 ∈ 𝑋 𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))))
25	24	adantl 482	. . . . . . . . 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 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
27	26	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑣 = (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
28	27	ac6sfi 8204	. . . . . . . . . . . . 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 752	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 1089	. . . . . . . . . . . . . . . . . . 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		frn 6053	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚:𝑢⟶𝑋 → ran 𝑚 ⊆ 𝑋)
33	31, 32	syl 17	. . . . . . . . . . . . . . . . . 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)
34	1	mopnuni 22246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
35	3, 5, 34	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 = ∪ 𝐽)
36	35	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋 = ∪ 𝐽)
37	36	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 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)
38	33, 37	sseqtrd 3641	. . . . . . . . . . . . . . . . 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)
39		fvex 6201	. . . . . . . . . . . . . . . . . . . 20 ⊢ (MetOpen‘𝐷) ∈ V
40	1, 39	eqeltri 2697	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ∈ V
41	40	uniex 6953	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝐽 ∈ V
42	41	elpw2 4828	. . . . . . . . . . . . . . . . 17 ⊢ (ran 𝑚 ∈ 𝒫 ∪ 𝐽 ↔ ran 𝑚 ⊆ ∪ 𝐽)
43	38, 42	sylibr 224	. . . . . . . . . . . . . . . 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)
44		simp2l 1087	. . . . . . . . . . . . . . . . 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)
45		ffn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚 Fn 𝑢)
46		dffn4 6121	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 Fn 𝑢 ↔ 𝑚:𝑢–onto→ran 𝑚)
47	45, 46	sylib 208	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚:𝑢⟶𝑋 → 𝑚:𝑢–onto→ran 𝑚)
48		fofi 8252	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ Fin ∧ 𝑚:𝑢–onto→ran 𝑚) → ran 𝑚 ∈ Fin)
49	47, 48	sylan2 491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ Fin ∧ 𝑚:𝑢⟶𝑋) → ran 𝑚 ∈ Fin)
50	44, 31, 49	syl2anc 693	. . . . . . . . . . . . . . . 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)
51	43, 50	elind 3798	. . . . . . . . . . . . . . 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))
52	26	eleq2d 2687	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑚‘𝑣) → (𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
53	52	rexrn 6361	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 Fn 𝑢 → (∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣 ∈ 𝑢 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
54		eliun 4524	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑦 ∈ ran 𝑚 𝑟 ∈ (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
55		eliun 4524	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))) ↔ ∃𝑣 ∈ 𝑢 𝑟 ∈ ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
56	53, 54, 55	3bitr4g 303	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 Fn 𝑢 → (𝑟 ∈ ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑟 ∈ ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛)))))
57	56	eqrdv 2620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 Fn 𝑢 → ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
58	31, 45, 57	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))))
59		simp3r 1090	. . . . . . . . . . . . . . . . 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))))
60		uniiun 4573	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑢 = ∪ 𝑣 ∈ 𝑢 𝑣
61		iuneq2 4537	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → ∪ 𝑣 ∈ 𝑢 𝑣 = ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
62	60, 61	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑣 ∈ 𝑢 𝑣 = ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))) → ∪ 𝑢 = ∪ 𝑣 ∈ 𝑢 ((𝑚‘𝑣)(ball‘𝐷)(1 / (2↑𝑛))))
63	59, 62	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))))
64		simp2r 1088	. . . . . . . . . . . . . . . 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)
65	58, 63, 64	3eqtr2rd 2663	. . . . . . . . . . . . . . 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))))
66		iuneq1 4534	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = ran 𝑚 → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
67	66	eqeq2d 2632	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ran 𝑚 → (𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑋 = ∪ 𝑦 ∈ ran 𝑚(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
68	67	rspcev 3309	. . . . . . . . . . . . . . 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))))
69	51, 65, 68	syl2anc 693	. . . . . . . . . . . . . 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))))
70	69	3expia 1267	. . . . . . . . . . . . 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)))))
71	70	adantrrr 761	. . . . . . . . . . . 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)))))
72	71	exlimdv 1861	. . . . . . . . . . 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)))))
73	30, 72	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))))
74	73	rexlimdvaa 3032	. . . . . . . . 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)))))
75	25, 74	syld 47	. . . . . . . 8 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑛 ∈ ℕ0) → (𝐷 ∈ (TotBnd‘𝑋) → ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
76	75	ralrimdva 2969	. . . . . . 7 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑛 ∈ ℕ0 ∃𝑡 ∈ (𝒫 ∪ 𝐽 ∩ Fin)𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
77	41	pwex 4848	. . . . . . . . 9 ⊢ 𝒫 ∪ 𝐽 ∈ V
78	77	inex1 4799	. . . . . . . 8 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ∈ V
79		nn0ennn 12778	. . . . . . . . 9 ⊢ ℕ0 ≈ ℕ
80		nnenom 12779	. . . . . . . . 9 ⊢ ℕ ≈ ω
81	79, 80	entri 8010	. . . . . . . 8 ⊢ ℕ0 ≈ ω
82		iuneq1 4534	. . . . . . . . 9 ⊢ (𝑡 = (𝑚‘𝑛) → ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))
83	82	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑡 = (𝑚‘𝑛) → (𝑋 = ∪ 𝑦 ∈ 𝑡 (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ↔ 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛)))))
84	78, 81, 83	axcc4 9261	. . . . . . 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)))))
85	76, 84	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))))))
86		elpwi 4168	. . . . . . . . . 10 ⊢ (𝑟 ∈ 𝒫 𝐽 → 𝑟 ⊆ 𝐽)
87		eqid 2622	. . . . . . . . . . . 12 ⊢ {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣} = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
88		eqid 2622	. . . . . . . . . . . 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})}
89		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
90		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝑚‘𝑛)(𝑦(ball‘𝐷)(1 / (2↑𝑛))))) → 𝐷 ∈ (CMet‘𝑋))
91	35	pweqd 4163	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
92	91	ineq1d 3813	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 𝑋 ∩ Fin) = (𝒫 ∪ 𝐽 ∩ Fin))
93	92	feq3d 6032	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin) ↔ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)))
94	93	biimpar 502	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶(𝒫 𝑋 ∩ Fin))
95	94	adantrr 753	. . . . . . . . . . . 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))
96		oveq1 6657	. . . . . . . . . . . . . . . . . . 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))
97	96	cbviunv 4559	. . . . . . . . . . . . . . . . . 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)
98		id 22	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) → 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin))
99		inss1 3833	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 ∪ 𝐽
100	99, 91	syl5sseqr 3654	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋)
101		fss 6056	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin) ∧ (𝒫 ∪ 𝐽 ∩ Fin) ⊆ 𝒫 𝑋) → 𝑚:ℕ0⟶𝒫 𝑋)
102	98, 100, 101	syl2anr 495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) → 𝑚:ℕ0⟶𝒫 𝑋)
103	102	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ∈ 𝒫 𝑋)
104	103	elpwid 4170	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) → (𝑚‘𝑛) ⊆ 𝑋)
105	104	sselda 3603	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → 𝑦 ∈ 𝑋)
106		simplr 792	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐷 ∈ (CMet‘𝑋) ∧ 𝑚:ℕ0⟶(𝒫 ∪ 𝐽 ∩ Fin)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑦 ∈ (𝑚‘𝑛)) → 𝑛 ∈ ℕ0)
107		oveq1 6657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑚))))
108		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
109	108	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (1 / (2↑𝑚)) = (1 / (2↑𝑛)))
110	109	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (𝑦(ball‘𝐷)(1 / (2↑𝑚))) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
111		ovex 6678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦(ball‘𝐷)(1 / (2↑𝑛))) ∈ V
112	107, 110, 89, 111	ovmpt2 6796	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑋 ∧ 𝑛 ∈ ℕ0) → (𝑦(𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))𝑛) = (𝑦(ball‘𝐷)(1 / (2↑𝑛))))
113	105, 106, 112	syl2anc 693	. . . . . . . . . . . . . . . . . . 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))))
114	113	iuneq2dv 4542	. . . . . . . . . . . . . . . . . 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))))
115	97, 114	syl5eq 2668	. . . . . . . . . . . . . . . . 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))))
116	115	eqeq2d 2632	. . . . . . . . . . . . . . . 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)))))
117	116	biimprd 238	. . . . . . . . . . . . . . 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)))
118	117	ralimdva 2962	. . . . . . . . . . . . . 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)))
119	118	impr 649	. . . . . . . . . . . . 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))
120		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝑚‘𝑛) = (𝑚‘𝑘))
121	120	iuneq1d 4545	. . . . . . . . . . . . . . . 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))
122		simpl 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 = 𝑘 ∧ 𝑡 ∈ (𝑚‘𝑘)) → 𝑛 = 𝑘)
123	122	oveq2d 6666	. . . . . . . . . . . . . . . . 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))
124	123	iuneq2dv 4542	. . . . . . . . . . . . . . . 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))
125	121, 124	eqtrd 2656	. . . . . . . . . . . . . . 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))
126	125	eqeq2d 2632	. . . . . . . . . . . . . 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)))
127	126	cbvralv 3171	. . . . . . . . . . . . 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))
128	119, 127	sylib 208	. . . . . . . . . . . 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))
129	1, 87, 88, 89, 90, 95, 128	heiborlem10 33619	. . . . . . . . . . 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)
130	129	exp32 631	. . . . . . . . . 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)))
131	86, 130	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)))
132	131	ralrimiv 2965	. . . . . . . 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))
133	132	ex 450	. . . . . . 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)))
134	133	exlimdv 1861	. . . . . 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)))
135	85, 134	syld 47	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → (𝐷 ∈ (TotBnd‘𝑋) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
136	135	imp 445	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣))
137		eqid 2622	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
138	137	iscmp 21191	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑣 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑣)))
139	8, 136, 138	sylanbrc 698	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → 𝐽 ∈ Comp)
140	4, 139	jca 554	. 2 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)) → (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp))
141	2, 140	impbii 199	1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520  {copab 4712  ran crn 5115   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  Fincfn 7955  1c1 9937   / cdiv 10684  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℝ⁺crp 11832  ↑cexp 12860  ∞Metcxmt 19731  Metcme 19732  ballcbl 19733  MetOpencmopn 19736  Topctop 20698  Compccmp 21189  CMetcms 23052  TotBndctotbnd 33565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lm 21033  df-haus 21119  df-cmp 21190  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cfil 23053  df-cau 23054  df-cmet 23055  df-totbnd 33567

This theorem is referenced by:  rrnheibor  33636