Theorem iscmet3 25191

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 25187	. . . 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 3121	. 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 12897	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
9		rphalfcl 12922	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ+ → (1 / 2) ∈ ℝ+)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ+
11		rpexpcl 13987	. . . . . . . . . 10 ⊢ (((1 / 2) ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → ((1 / 2)↑𝑘) ∈ ℝ+)
12	10, 11	mpan 690	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ → ((1 / 2)↑𝑘) ∈ ℝ+)
13		cfili 25166	. . . . . . . . 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 3121	. . . . . . 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 3440	. . . . . . . 8 ⊢ 𝑔 ∈ V
17		znnen 16121	. . . . . . . . 9 ⊢ ℤ ≈ ℕ
18		nnenom 13887	. . . . . . . . 9 ⊢ ℕ ≈ ω
19	17, 18	entri 8933	. . . . . . . 8 ⊢ ℤ ≈ ω
20		raleq 3286	. . . . . . . . 9 ⊢ (𝑡 = (𝑠‘𝑘) → (∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
21	20	raleqbi1dv 3301	. . . . . . . 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 10333	. . . . . . 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 13871	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
28		endom 8904	. . . . . . . . . . 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 3911	. . . . . . . . . . . . . . 15 ⊢ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)}
31		fzn0 13441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀...𝑘) ≠ ∅ ↔ 𝑘 ∈ (ℤ≥‘𝑀))
32	31	biimpri 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...𝑘) ≠ ∅)
33	32, 26	eleq2s 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑍 → (𝑀...𝑘) ≠ ∅)
34		metxmet 24220	. . . . . . . . . . . . . . . . . . . . . . . 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 25165	. . . . . . . . . . . . . . . . . . . . . 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 13427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (𝑀...𝑘) → 𝑛 ∈ ℤ)
42		ffvelcdm 7015	. . . . . . . . . . . . . . . . . . . . . 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 23737	. . . . . . . . . . . . . . . . . . . . 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 3121	. . . . . . . . . . . . . . . . . . 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 4446	. . . . . . . . . . . . . . . . . . 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 5005	. . . . . . . . . . . . . . . . . 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 6857	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ dom Met)
53		fvi 6899	. . . . . . . . . . . . . . . . . 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 3973	. . . . . . . . . . . . . . . 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 4174	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . 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 3121	. . . . . . . . . . . . . . . . . 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 13880	. . . . . . . . . . . . . . . . 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 9307	. . . . . . . . . . . . . . . . 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 23744	. . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . 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 23735	. . . . . . . . . . . . . . 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 2992	. . . . . . . . . . . . 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 4340	. . . . . . . . . . . . 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 3121	. . . . . . . . . . 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 6835	. . . . . . . . . . 11 ⊢ ( I ‘𝑋) ∈ V
77		eleq1 2816	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)))
78		fvex 6835	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑘) ∈ V
79		eliin 4946	. . . . . . . . . . . . 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 10335	. . . . . . . . . 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 3045	. . . . . . . . . . . . 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 1873	. . . . . . . . . . . . 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 6632	. . . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝑠‘𝑘) = (𝑠‘𝑖))
96		oveq2 7357	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((1 / 2)↑𝑘) = ((1 / 2)↑𝑖))
97	96	breq2d 5104	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ (𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
98	95, 97	raleqbidv 3309	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
99	95, 98	raleqbidv 3309	. . . . . . . . . . . . . . . . 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 3207	. . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝑠‘𝑛) = (𝑠‘𝑗))
104	103	eleq2d 2814	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ (𝑠‘𝑗)))
105	104	cbvralvw 3207	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗))
106		oveq2 7357	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑀...𝑘) = (𝑀...𝑖))
107		fveq2 6822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → (𝑓‘𝑘) = (𝑓‘𝑖))
108	107	eleq1d 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ (𝑓‘𝑖) ∈ (𝑠‘𝑗)))
109	106, 108	raleqbidv 3309	. . . . . . . . . . . . . . . . . 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 3207	. . . . . . . . . . . . . . . 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 25189	. . . . . . . . . . . . . . . 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 25190	. . . . . . . . . . . . . 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 1933	. . . . . . . . . . . 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 1933	. . . . . 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 3121	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅)
131	1	iscmet 25182	. . . 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 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3394  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  ∩ ciin 4942   class class class wbr 5092   I cid 5513  dom cdm 5619  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  ωcom 7799   ≈ cen 8869   ≼ cdom 8870  Fincfn 8872  ficfi 9300  1c1 11010   < clt 11149   / cdiv 11777  ℕcn 12128  2c2 12183  ℤcz 12471  ℤ_≥cuz 12735  ℝ⁺crp 12893  ...cfz 13410  ↑cexp 13968  ∞Metcxmet 21246  Metcmet 21247  MetOpencmopn 21251  ⇝_𝑡clm 23111  Filcfil 23730   fLim cflim 23819  CauFilccfil 25150  Cauccau 25151  CMetccmet 25152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cc 10329  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-card 9835  df-acn 9838  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ico 13254  df-fz 13411  df-fl 13696  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-rest 17326  df-topgen 17347  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-fbas 21258  df-fg 21259  df-top 22779  df-topon 22796  df-bases 22831  df-ntr 22905  df-nei 22983  df-lm 23114  df-fil 23731  df-fm 23823  df-flim 23824  df-flf 23825  df-cfil 25153  df-cau 25154  df-cmet 25155

This theorem is referenced by:  iscmet2  25192  iscmet3i  25210  heibor1  37794  rrncms  37817