Theorem iscmet3 25249

Description: The property "𝐷 is a complete metric" expressed in terms of functions on ℕ (or any other upper integer set). Thus, we only have to look at functions on ℕ, and not all possible Cauchy filters, to determine completeness. (The proof uses countable choice.) (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 5-May-2014.)

Hypotheses

Ref	Expression
iscmet3.1	⊢ 𝑍 = (ℤ≥‘𝑀)
iscmet3.2	⊢ 𝐽 = (MetOpen‘𝐷)
iscmet3.3	⊢ (𝜑 → 𝑀 ∈ ℤ)
iscmet3.4	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))

Assertion

Ref	Expression
iscmet3	⊢ (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))

Distinct variable groups:   𝐷,𝑓   𝑓,𝑋   𝑓,𝐽   𝑓,𝑍   𝑓,𝑀   𝜑,𝑓

Proof of Theorem iscmet3

Dummy variables 𝑔 𝑖 𝑗 𝑘 𝑛 𝑠 𝑡 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscmet3.2	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷)
2	1	cmetcau 25245	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑓 ∈ (Cau‘𝐷)) → 𝑓 ∈ dom (⇝𝑡‘𝐽))
3	2	a1d 25	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑓 ∈ (Cau‘𝐷)) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)))
4	3	ralrimiva 3128	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) → ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)))
5		iscmet3.4	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (Met‘𝑋))
7		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (CauFil‘𝐷))
8		1rp 12909	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
9		rphalfcl 12934	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ+ → (1 / 2) ∈ ℝ+)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ+
11		rpexpcl 14003	. . . . . . . . . 10 ⊢ (((1 / 2) ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → ((1 / 2)↑𝑘) ∈ ℝ+)
12	10, 11	mpan 690	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ → ((1 / 2)↑𝑘) ∈ ℝ+)
13		cfili 25224	. . . . . . . . 9 ⊢ ((𝑔 ∈ (CauFil‘𝐷) ∧ ((1 / 2)↑𝑘) ∈ ℝ+) → ∃𝑡 ∈ 𝑔 ∀𝑢 ∈ 𝑡 ∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
14	7, 12, 13	syl2an 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) ∧ 𝑘 ∈ ℤ) → ∃𝑡 ∈ 𝑔 ∀𝑢 ∈ 𝑡 ∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
15	14	ralrimiva 3128	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → ∀𝑘 ∈ ℤ ∃𝑡 ∈ 𝑔 ∀𝑢 ∈ 𝑡 ∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
16		vex 3444	. . . . . . . 8 ⊢ 𝑔 ∈ V
17		znnen 16137	. . . . . . . . 9 ⊢ ℤ ≈ ℕ
18		nnenom 13903	. . . . . . . . 9 ⊢ ℕ ≈ ω
19	17, 18	entri 8945	. . . . . . . 8 ⊢ ℤ ≈ ω
20		raleq 3293	. . . . . . . . 9 ⊢ (𝑡 = (𝑠‘𝑘) → (∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
21	20	raleqbi1dv 3308	. . . . . . . 8 ⊢ (𝑡 = (𝑠‘𝑘) → (∀𝑢 ∈ 𝑡 ∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
22	16, 19, 21	axcc4 10349	. . . . . . 7 ⊢ (∀𝑘 ∈ ℤ ∃𝑡 ∈ 𝑔 ∀𝑢 ∈ 𝑡 ∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) → ∃𝑠(𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
23	15, 22	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → ∃𝑠(𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
24		iscmet3.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
25	24	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → 𝑀 ∈ ℤ)
26		iscmet3.1	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ≥‘𝑀)
27	26	uzenom 13887	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
28		endom 8916	. . . . . . . . . . 11 ⊢ (𝑍 ≈ ω → 𝑍 ≼ ω)
29	25, 27, 28	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → 𝑍 ≼ ω)
30		dfin5 3909	. . . . . . . . . . . . . . 15 ⊢ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)}
31		fzn0 13454	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀...𝑘) ≠ ∅ ↔ 𝑘 ∈ (ℤ≥‘𝑀))
32	31	biimpri 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...𝑘) ≠ ∅)
33	32, 26	eleq2s 2854	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑍 → (𝑀...𝑘) ≠ ∅)
34		metxmet 24278	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
35	5, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
36	35	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (∞Met‘𝑋))
37		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔) → 𝑔 ∈ (CauFil‘𝐷))
38		cfilfil 25223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (Fil‘𝑋))
39	36, 37, 38	syl2an 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑔 ∈ (Fil‘𝑋))
40		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑠:ℤ⟶𝑔)
41		elfzelz 13440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (𝑀...𝑘) → 𝑛 ∈ ℤ)
42		ffvelcdm 7026	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠:ℤ⟶𝑔 ∧ 𝑛 ∈ ℤ) → (𝑠‘𝑛) ∈ 𝑔)
43	40, 41, 42	syl2an 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑠‘𝑛) ∈ 𝑔)
44		filelss 23796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ (𝑠‘𝑛) ∈ 𝑔) → (𝑠‘𝑛) ⊆ 𝑋)
45	39, 43, 44	syl2an2r 685	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑠‘𝑛) ⊆ 𝑋)
46	45	ralrimiva 3128	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
47		r19.2z 4452	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀...𝑘) ≠ ∅ ∧ ∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋) → ∃𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
48	33, 46, 47	syl2anr 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ∃𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
49		iinss 5012	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋 → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
50	48, 49	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
51	6	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → 𝐷 ∈ (Met‘𝑋))
52		elfvdm 6868	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ dom Met)
53		fvi 6910	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ dom Met → ( I ‘𝑋) = 𝑋)
54	51, 52, 53	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ( I ‘𝑋) = 𝑋)
55	50, 54	sseqtrrd 3971	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ ( I ‘𝑋))
56		sseqin2 4175	. . . . . . . . . . . . . . . 16 ⊢ (∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ ( I ‘𝑋) ↔ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
57	55, 56	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
58	30, 57	eqtr3id 2785	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)} = ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
59	39	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → 𝑔 ∈ (Fil‘𝑋))
60	43	ralrimiva 3128	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔)
61	60	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔)
62	33	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → (𝑀...𝑘) ≠ ∅)
63		fzfid 13896	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → (𝑀...𝑘) ∈ Fin)
64		iinfi 9320	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ (∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔 ∧ (𝑀...𝑘) ≠ ∅ ∧ (𝑀...𝑘) ∈ Fin)) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ (fi‘𝑔))
65	59, 61, 62, 63, 64	syl13anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ (fi‘𝑔))
66		filfi 23803	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ (Fil‘𝑋) → (fi‘𝑔) = 𝑔)
67	59, 66	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → (fi‘𝑔) = 𝑔)
68	65, 67	eleqtrd 2838	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔)
69		fileln0 23794	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ≠ ∅)
70	39, 68, 69	syl2an2r 685	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ≠ ∅)
71	58, 70	eqnetrd 2999	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)} ≠ ∅)
72		rabn0 4341	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)} ≠ ∅ ↔ ∃𝑥 ∈ ( I ‘𝑋)𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
73	71, 72	sylib 218	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → ∃𝑥 ∈ ( I ‘𝑋)𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
74	73	ralrimiva 3128	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑘 ∈ 𝑍 ∃𝑥 ∈ ( I ‘𝑋)𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
75	74	adantrrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → ∀𝑘 ∈ 𝑍 ∃𝑥 ∈ ( I ‘𝑋)𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
76		fvex 6847	. . . . . . . . . . 11 ⊢ ( I ‘𝑋) ∈ V
77		eleq1 2824	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)))
78		fvex 6847	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑘) ∈ V
79		eliin 4951	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑘) ∈ V → ((𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))
80	78, 79	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))
81	77, 80	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))
82	76, 81	axcc4dom 10351	. . . . . . . . . 10 ⊢ ((𝑍 ≼ ω ∧ ∀𝑘 ∈ 𝑍 ∃𝑥 ∈ ( I ‘𝑋)𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) → ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))
83	29, 75, 82	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))
84		df-ral 3052	. . . . . . . . . . . . 13 ⊢ (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)) ↔ ∀𝑓(𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))
85		19.29 1874	. . . . . . . . . . . . 13 ⊢ ((∀𝑓(𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))) → ∃𝑓((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))))
86	84, 85	sylanb 581	. . . . . . . . . . . 12 ⊢ ((∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)) ∧ ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))) → ∃𝑓((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))))
87	24	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝑀 ∈ ℤ)
88	5	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝐷 ∈ (Met‘𝑋))
89		simprrl 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝑓:𝑍⟶( I ‘𝑋))
90		feq3 6642	. . . . . . . . . . . . . . . . 17 ⊢ (( I ‘𝑋) = 𝑋 → (𝑓:𝑍⟶( I ‘𝑋) ↔ 𝑓:𝑍⟶𝑋))
91	88, 52, 53, 90	4syl 19	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → (𝑓:𝑍⟶( I ‘𝑋) ↔ 𝑓:𝑍⟶𝑋))
92	89, 91	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝑓:𝑍⟶𝑋)
93		simplrr 777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
94	93	simprd 495	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
95		fveq2 6834	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝑠‘𝑘) = (𝑠‘𝑖))
96		oveq2 7366	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((1 / 2)↑𝑘) = ((1 / 2)↑𝑖))
97	96	breq2d 5110	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ (𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
98	95, 97	raleqbidv 3316	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
99	95, 98	raleqbidv 3316	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑢 ∈ (𝑠‘𝑖)∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
100	99	cbvralvw 3214	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑖 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑖)∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖))
101	94, 100	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → ∀𝑖 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑖)∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖))
102		simprrr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))
103		fveq2 6834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝑠‘𝑛) = (𝑠‘𝑗))
104	103	eleq2d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ (𝑠‘𝑗)))
105	104	cbvralvw 3214	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗))
106		oveq2 7366	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑀...𝑘) = (𝑀...𝑖))
107		fveq2 6834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → (𝑓‘𝑘) = (𝑓‘𝑖))
108	107	eleq1d 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ (𝑓‘𝑖) ∈ (𝑠‘𝑗)))
109	106, 108	raleqbidv 3316	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗)))
110	105, 109	bitrid 283	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗)))
111	110	cbvralvw 3214	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑖 ∈ 𝑍 ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗))
112	102, 111	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → ∀𝑖 ∈ 𝑍 ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗))
113	88, 34	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝐷 ∈ (∞Met‘𝑋))
114		simplrl 776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝑔 ∈ (CauFil‘𝐷))
115	113, 114, 38	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝑔 ∈ (Fil‘𝑋))
116	93	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝑠:ℤ⟶𝑔)
117	26, 1, 87, 88, 92, 101, 112	iscmet3lem1 25247	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝑓 ∈ (Cau‘𝐷))
118		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → (𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))
119	117, 92, 118	mp2d 49	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → 𝑓 ∈ dom (⇝𝑡‘𝐽))
120	26, 1, 87, 88, 92, 101, 112, 115, 116, 119	iscmet3lem2 25248	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))) → (𝐽 fLim 𝑔) ≠ ∅)
121	120	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → (((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))) → (𝐽 fLim 𝑔) ≠ ∅))
122	121	exlimdv 1934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → (∃𝑓((𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))) → (𝐽 fLim 𝑔) ≠ ∅))
123	86, 122	syl5 34	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → ((∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)) ∧ ∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))) → (𝐽 fLim 𝑔) ≠ ∅))
124	123	expdimp 452	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → (∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)) → (𝐽 fLim 𝑔) ≠ ∅))
125	124	an32s 652	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → (∃𝑓(𝑓:𝑍⟶( I ‘𝑋) ∧ ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)) → (𝐽 fLim 𝑔) ≠ ∅))
126	83, 125	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ (𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))) → (𝐽 fLim 𝑔) ≠ ∅)
127	126	expr 456	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → ((𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)) → (𝐽 fLim 𝑔) ≠ ∅))
128	127	exlimdv 1934	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → (∃𝑠(𝑠:ℤ⟶𝑔 ∧ ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)) → (𝐽 fLim 𝑔) ≠ ∅))
129	23, 128	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑔) ≠ ∅)
130	129	ralrimiva 3128	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅)
131	1	iscmet 25240	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅))
132	6, 130, 131	sylanbrc 583	. . 3 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (CMet‘𝑋))
133	132	ex 412	. 2 ⊢ (𝜑 → (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)) → 𝐷 ∈ (CMet‘𝑋)))
134	4, 133	impbid2 226	1 ⊢ (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  ∩ ciin 4947   class class class wbr 5098   I cid 5518  dom cdm 5624  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  ωcom 7808   ≈ cen 8880   ≼ cdom 8881  Fincfn 8883  ficfi 9313  1c1 11027   < clt 11166   / cdiv 11794  ℕcn 12145  2c2 12200  ℤcz 12488  ℤ_≥cuz 12751  ℝ⁺crp 12905  ...cfz 13423  ↑cexp 13984  ∞Metcxmet 21294  Metcmet 21295  MetOpencmopn 21299  ⇝_𝑡clm 23170  Filcfil 23789   fLim cflim 23878  CauFilccfil 25208  Cauccau 25209  CMetccmet 25210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cc 10345  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-omul 8402  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-acn 9854  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-q 12862  df-rp 12906  df-xneg 13026  df-xadd 13027  df-xmul 13028  df-ico 13267  df-fz 13424  df-fl 13712  df-seq 13925  df-exp 13985  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-rlim 15412  df-rest 17342  df-topgen 17363  df-psmet 21301  df-xmet 21302  df-met 21303  df-bl 21304  df-mopn 21305  df-fbas 21306  df-fg 21307  df-top 22838  df-topon 22855  df-bases 22890  df-ntr 22964  df-nei 23042  df-lm 23173  df-fil 23790  df-fm 23882  df-flim 23883  df-flf 23884  df-cfil 25211  df-cau 25212  df-cmet 25213

This theorem is referenced by:  iscmet2  25250  iscmet3i  25268  heibor1  38011  rrncms  38034