Theorem iscmet3 24457

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 24453	. . . 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 3103	. 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 481	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (Met‘𝑋))
7		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (CauFil‘𝐷))
8		1rp 12734	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
9		rphalfcl 12757	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ+ → (1 / 2) ∈ ℝ+)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ+
11		rpexpcl 13801	. . . . . . . . . 10 ⊢ (((1 / 2) ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → ((1 / 2)↑𝑘) ∈ ℝ+)
12	10, 11	mpan 687	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ → ((1 / 2)↑𝑘) ∈ ℝ+)
13		cfili 24432	. . . . . . . . 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 3103	. . . . . . 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 3436	. . . . . . . 8 ⊢ 𝑔 ∈ V
17		znnen 15921	. . . . . . . . 9 ⊢ ℤ ≈ ℕ
18		nnenom 13700	. . . . . . . . 9 ⊢ ℕ ≈ ω
19	17, 18	entri 8794	. . . . . . . 8 ⊢ ℤ ≈ ω
20		raleq 3342	. . . . . . . . 9 ⊢ (𝑡 = (𝑠‘𝑘) → (∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
21	20	raleqbi1dv 3340	. . . . . . . 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 10195	. . . . . . 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 723	. . . . . . . . . . 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 13684	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
28		endom 8767	. . . . . . . . . . 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 3895	. . . . . . . . . . . . . . 15 ⊢ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)}
31		fzn0 13270	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀...𝑘) ≠ ∅ ↔ 𝑘 ∈ (ℤ≥‘𝑀))
32	31	biimpri 227	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...𝑘) ≠ ∅)
33	32, 26	eleq2s 2857	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑍 → (𝑀...𝑘) ≠ ∅)
34		metxmet 23487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
35	5, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
36	35	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (∞Met‘𝑋))
37		simpl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔) → 𝑔 ∈ (CauFil‘𝐷))
38		cfilfil 24431	. . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑠:ℤ⟶𝑔)
41		elfzelz 13256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (𝑀...𝑘) → 𝑛 ∈ ℤ)
42		ffvelrn 6959	. . . . . . . . . . . . . . . . . . . . . 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 23003	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ (𝑠‘𝑛) ∈ 𝑔) → (𝑠‘𝑛) ⊆ 𝑋)
45	39, 43, 44	syl2an2r 682	. . . . . . . . . . . . . . . . . . . 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 3103	. . . . . . . . . . . . . . . . . . 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 4425	. . . . . . . . . . . . . . . . . . 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 4986	. . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . 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 6806	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ dom Met)
53		fvi 6844	. . . . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . . . . 16 ⊢ (∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ ( I ‘𝑋) ↔ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
57	55, 56	sylib 217	. . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 3103	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → (𝑀...𝑘) ≠ ∅)
63		fzfid 13693	. . . . . . . . . . . . . . . . 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 9176	. . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . 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 23010	. . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . 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 23001	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ≠ ∅)
70	39, 68, 69	syl2an2r 682	. . . . . . . . . . . . . 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 3011	. . . . . . . . . . . . 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 4319	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)} ≠ ∅ ↔ ∃𝑥 ∈ ( I ‘𝑋)𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
73	71, 72	sylib 217	. . . . . . . . . . . 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 3103	. . . . . . . . . . 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 722	. . . . . . . . . 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 6787	. . . . . . . . . . 11 ⊢ ( I ‘𝑋) ∈ V
77		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)))
78		fvex 6787	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑘) ∈ V
79		eliin 4929	. . . . . . . . . . . . 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 10197	. . . . . . . . . 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 3069	. . . . . . . . . . . . 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 1876	. . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . 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 6583	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝑠‘𝑘) = (𝑠‘𝑖))
96		oveq2 7283	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((1 / 2)↑𝑘) = ((1 / 2)↑𝑖))
97	96	breq2d 5086	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ (𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
98	95, 97	raleqbidv 3336	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
99	95, 98	raleqbidv 3336	. . . . . . . . . . . . . . . . 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 3383	. . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝑠‘𝑛) = (𝑠‘𝑗))
104	103	eleq2d 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ (𝑠‘𝑗)))
105	104	cbvralvw 3383	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗))
106		oveq2 7283	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑀...𝑘) = (𝑀...𝑖))
107		fveq2 6774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → (𝑓‘𝑘) = (𝑓‘𝑖))
108	107	eleq1d 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ (𝑓‘𝑖) ∈ (𝑠‘𝑗)))
109	106, 108	raleqbidv 3336	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗)))
110	105, 109	bitrid 282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗)))
111	110	cbvralvw 3383	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑖 ∈ 𝑍 ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗))
112	102, 111	sylib 217	. . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . 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 24455	. . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 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 24456	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 1936	. . . . . . . . . . . 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 453	. . . . . . . . . 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 649	. . . . . . . . 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 457	. . . . . . 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 1936	. . . . . 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 3103	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅)
131	1	iscmet 24448	. . . 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 413	. 2 ⊢ (𝜑 → (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)) → 𝐷 ∈ (CMet‘𝑋)))
134	4, 133	impbid2 225	1 ⊢ (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ∩ ciin 4925   class class class wbr 5074   I cid 5488  dom cdm 5589  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ωcom 7712   ≈ cen 8730   ≼ cdom 8731  Fincfn 8733  ficfi 9169  1c1 10872   < clt 11009   / cdiv 11632  ℕcn 11973  2c2 12028  ℤcz 12319  ℤ_≥cuz 12582  ℝ⁺crp 12730  ...cfz 13239  ↑cexp 13782  ∞Metcxmet 20582  Metcmet 20583  MetOpencmopn 20587  ⇝_𝑡clm 22377  Filcfil 22996   fLim cflim 23085  CauFilccfil 24416  Cauccau 24417  CMetccmet 24418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-acn 9700  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ico 13085  df-fz 13240  df-fl 13512  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-top 22043  df-topon 22060  df-bases 22096  df-ntr 22171  df-nei 22249  df-lm 22380  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-cfil 24419  df-cau 24420  df-cmet 24421

This theorem is referenced by:  iscmet2  24458  iscmet3i  24476  heibor1  35968  rrncms  35991