Theorem iscmet3 25623

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 25619	. . . 4 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑓 ∈ (Cau‘𝐷)) → 𝑓 ∈ dom (⇝𝑡‘𝐽))
3	2	a1d 26	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑓 ∈ (Cau‘𝐷)) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)))
4	3	ralrimiva 3154	. 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 486	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (Met‘𝑋))
7		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (CauFil‘𝐷))
8		1rp 13131	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
9		rphalfcl 13156	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ+ → (1 / 2) ∈ ℝ+)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ+
11		rpexpcl 14230	. . . . . . . . . 10 ⊢ (((1 / 2) ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → ((1 / 2)↑𝑘) ∈ ℝ+)
12	10, 11	mpan 703	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ → ((1 / 2)↑𝑘) ∈ ℝ+)
13		cfili 25598	. . . . . . . . 9 ⊢ ((𝑔 ∈ (CauFil‘𝐷) ∧ ((1 / 2)↑𝑘) ∈ ℝ+) → ∃𝑡 ∈ 𝑔 ∀𝑢 ∈ 𝑡 ∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
14	7, 12, 13	syl2an 608	. . . . . . . 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 3154	. . . . . . 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 3454	. . . . . . . 8 ⊢ 𝑔 ∈ V
17		znnen 16389	. . . . . . . . 9 ⊢ ℤ ≈ ℕ
18		nnenom 14130	. . . . . . . . 9 ⊢ ℕ ≈ ω
19	17, 18	entri 9043	. . . . . . . 8 ⊢ ℤ ≈ ω
20		raleq 3316	. . . . . . . . 9 ⊢ (𝑡 = (𝑠‘𝑘) → (∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
21	20	raleqbi1dv 3329	. . . . . . . 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 10524	. . . . . . 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 18	. . . . . 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 739	. . . . . . . . . . 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 14114	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
28		endom 9014	. . . . . . . . . . 11 ⊢ (𝑍 ≈ ω → 𝑍 ≼ ω)
29	25, 27, 28	3syl 19	. . . . . . . . . 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 4091	. . . . . . . . . . . . . . 15 ⊢ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)}
31		fzn0 13678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀...𝑘) ≠ ∅ ↔ 𝑘 ∈ (ℤ≥‘𝑀))
32	31	biimpri 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...𝑘) ≠ ∅)
33	32, 26	eleq2s 2878	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑍 → (𝑀...𝑘) ≠ ∅)
34		metxmet 24662	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
35	5, 34	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
36	35	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (∞Met‘𝑋))
37		simpl 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔) → 𝑔 ∈ (CauFil‘𝐷))
38		cfilfil 25597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (Fil‘𝑋))
39	36, 37, 38	syl2an 608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑔 ∈ (Fil‘𝑋))
40		simprr 785	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑠:ℤ⟶𝑔)
41		elfzelz 13663	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (𝑀...𝑘) → 𝑛 ∈ ℤ)
42		ffvelcdm 7086	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠:ℤ⟶𝑔 ∧ 𝑛 ∈ ℤ) → (𝑠‘𝑛) ∈ 𝑔)
43	40, 41, 42	syl2an 608	. . . . . . . . . . . . . . . . . . . . 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 24180	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ (𝑠‘𝑛) ∈ 𝑔) → (𝑠‘𝑛) ⊆ 𝑋)
45	39, 43, 44	syl2an2r 698	. . . . . . . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . . . . . . . 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 4454	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀...𝑘) ≠ ∅ ∧ ∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋) → ∃𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
48	33, 46, 47	syl2anr 609	. . . . . . . . . . . . . . . . . 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 5014	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋 → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
50	48, 49	syl 18	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ dom Met)
53		fvi 6966	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ dom Met → ( I ‘𝑋) = 𝑋)
54	51, 52, 53	3syl 19	. . . . . . . . . . . . . . . . 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 3946	. . . . . . . . . . . . . . . 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 4119	. . . . . . . . . . . . . . . 16 ⊢ (∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ ( I ‘𝑋) ↔ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
57	55, 56	sylib 221	. . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → (𝑀...𝑘) ≠ ∅)
63		fzfid 14123	. . . . . . . . . . . . . . . . 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 9416	. . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . 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 24187	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ (Fil‘𝑋) → (fi‘𝑔) = 𝑔)
67	59, 66	syl 18	. . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . 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 24178	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ≠ ∅)
70	39, 68, 69	syl2an2r 698	. . . . . . . . . . . . . 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 3022	. . . . . . . . . . . . 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 4338	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)} ≠ ∅ ↔ ∃𝑥 ∈ ( I ‘𝑋)𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
73	71, 72	sylib 221	. . . . . . . . . . . 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 3154	. . . . . . . . . . 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 738	. . . . . . . . . 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 6903	. . . . . . . . . . 11 ⊢ ( I ‘𝑋) ∈ V
77		eleq1 2848	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)))
78		fvex 6903	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑘) ∈ V
79		eliin 4955	. . . . . . . . . . . . 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 290	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))
82	76, 81	axcc4dom 10526	. . . . . . . . . 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 596	. . . . . . . . 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 3077	. . . . . . . . . . . . 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 1905	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 739	. . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . 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 6694	. . . . . . . . . . . . . . . . 17 ⊢ (( I ‘𝑋) = 𝑋 → (𝑓:𝑍⟶( I ‘𝑋) ↔ 𝑓:𝑍⟶𝑋))
91	88, 52, 53, 90	4syl 20	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝑠‘𝑘) = (𝑠‘𝑖))
96		oveq2 7432	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((1 / 2)↑𝑘) = ((1 / 2)↑𝑖))
97	96	breq2d 5114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ (𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
98	95, 97	raleqbidv 3334	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
99	95, 98	raleqbidv 3334	. . . . . . . . . . . . . . . . 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 3240	. . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝑠‘𝑛) = (𝑠‘𝑗))
104	103	eleq2d 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ (𝑠‘𝑗)))
105	104	cbvralvw 3240	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗))
106		oveq2 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑀...𝑘) = (𝑀...𝑖))
107		fveq2 6890	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → (𝑓‘𝑘) = (𝑓‘𝑖))
108	107	eleq1d 2845	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ (𝑓‘𝑖) ∈ (𝑠‘𝑗)))
109	106, 108	raleqbidv 3334	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗)))
110	105, 109	bitrid 286	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗)))
111	110	cbvralvw 3240	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑖 ∈ 𝑍 ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗))
112	102, 111	sylib 221	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . 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 25621	. . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . 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 50	. . . . . . . . . . . . . . 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 25622	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 1965	. . . . . . . . . . . 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 35	. . . . . . . . . . 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 458	. . . . . . . . . 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 665	. . . . . . . . 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 16	. . . . . . . 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 462	. . . . . . 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 1965	. . . . . 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 16	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑔) ≠ ∅)
130	129	ralrimiva 3154	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅)
131	1	iscmet 25614	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅))
132	6, 130, 131	sylanbrc 595	. . 3 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (CMet‘𝑋))
133	132	ex 418	. 2 ⊢ (𝜑 → (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)) → 𝐷 ∈ (CMet‘𝑋)))
134	4, 133	impbid2 229	1 ⊢ (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088  ∅c0 4277  ∩ ciin 4951   class class class wbr 5102   I cid 5545  dom cdm 5651  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  ωcom 7881   ≈ cen 8978   ≼ cdom 8979  Fincfn 8981  ficfi 9409  1c1 11208   < clt 11350   / cdiv 11980  ℕcn 12342  2c2 12404  ℤcz 12700  ℤ_≥cuz 12972  ℝ⁺crp 13127  ...cfz 13646  ↑cexp 14211  ∞Metcxmet 21672  Metcmet 21673  MetOpencmopn 21677  ⇝_𝑡clm 23553  Filcfil 24173   fLim cflim 24262  CauFilccfil 25582  Cauccau 25583  CMetccmet 25584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-cc 10520  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-2o 8485  df-oadd 8488  df-omul 8489  df-er 8725  df-map 8857  df-pm 8858  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-fi 9410  df-sup 9441  df-inf 9442  df-oi 9511  df-card 10027  df-acn 10030  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-3 12413  df-n0 12614  df-z 12701  df-uz 12973  df-q 13083  df-rp 13128  df-xneg 13248  df-xadd 13249  df-xmul 13250  df-ico 13489  df-fz 13647  df-fl 13939  df-seq 14152  df-exp 14212  df-cj 15275  df-re 15276  df-im 15277  df-sqrt 15411  df-abs 15412  df-clim 15664  df-rlim 15665  df-rest 17602  df-topgen 17623  df-psmet 21679  df-xmet 21680  df-met 21681  df-bl 21682  df-mopn 21683  df-fbas 21684  df-fg 21685  df-top 23221  df-topon 23238  df-bases 23273  df-ntr 23347  df-nei 23425  df-lm 23556  df-fil 24174  df-fm 24266  df-flim 24267  df-flf 24268  df-cfil 25585  df-cau 25586  df-cmet 25587

This theorem is used by:  iscmet2  25624  iscmet3i  25642  heibor1  38881  rrncms  38904