Theorem iscmet3 23309

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 23305	. . . 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 3115	. 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 466	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (Met‘𝑋))
7		simpr 471	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (CauFil‘𝐷))
8		1rp 12038	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
9		rphalfcl 12060	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ+ → (1 / 2) ∈ ℝ+)
10	8, 9	ax-mp 5	. . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ+
11		rpexpcl 13085	. . . . . . . . . 10 ⊢ (((1 / 2) ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → ((1 / 2)↑𝑘) ∈ ℝ+)
12	10, 11	mpan 670	. . . . . . . . 9 ⊢ (𝑘 ∈ ℤ → ((1 / 2)↑𝑘) ∈ ℝ+)
13		cfili 23284	. . . . . . . . 9 ⊢ ((𝑔 ∈ (CauFil‘𝐷) ∧ ((1 / 2)↑𝑘) ∈ ℝ+) → ∃𝑡 ∈ 𝑔 ∀𝑢 ∈ 𝑡 ∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
14	7, 12, 13	syl2an 583	. . . . . . . 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 3115	. . . . . . 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 3354	. . . . . . . 8 ⊢ 𝑔 ∈ V
17		znnen 15146	. . . . . . . . 9 ⊢ ℤ ≈ ℕ
18		nnenom 12986	. . . . . . . . 9 ⊢ ℕ ≈ ω
19	17, 18	entri 8166	. . . . . . . 8 ⊢ ℤ ≈ ω
20		raleq 3287	. . . . . . . . 9 ⊢ (𝑡 = (𝑠‘𝑘) → (∀𝑣 ∈ 𝑡 (𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
21	20	raleqbi1dv 3295	. . . . . . . 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 9466	. . . . . . 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 705	. . . . . . . . . . 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 12970	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω)
28		endom 8139	. . . . . . . . . . 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 3731	. . . . . . . . . . . . . . 15 ⊢ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = {𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)}
31		fzn0 12561	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀...𝑘) ≠ ∅ ↔ 𝑘 ∈ (ℤ≥‘𝑀))
32	31	biimpri 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑀...𝑘) ≠ ∅)
33	32, 26	eleq2s 2868	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝑍 → (𝑀...𝑘) ≠ ∅)
34		simprr 756	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑠:ℤ⟶𝑔)
35		elfzelz 12548	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (𝑀...𝑘) → 𝑛 ∈ ℤ)
36		ffvelrn 6502	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠:ℤ⟶𝑔 ∧ 𝑛 ∈ ℤ) → (𝑠‘𝑛) ∈ 𝑔)
37	34, 35, 36	syl2an 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑠‘𝑛) ∈ 𝑔)
38		metxmet 22358	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
39	5, 38	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
40	39	adantr 466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (∞Met‘𝑋))
41		simpl 468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔) → 𝑔 ∈ (CauFil‘𝐷))
42		cfilfil 23283	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑔 ∈ (CauFil‘𝐷)) → 𝑔 ∈ (Fil‘𝑋))
43	40, 41, 42	syl2an 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → 𝑔 ∈ (Fil‘𝑋))
44		filelss 21875	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ (𝑠‘𝑛) ∈ 𝑔) → (𝑠‘𝑛) ⊆ 𝑋)
45	43, 44	sylan 569	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ (𝑠‘𝑛) ∈ 𝑔) → (𝑠‘𝑛) ⊆ 𝑋)
46	37, 45	syldan 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑛 ∈ (𝑀...𝑘)) → (𝑠‘𝑛) ⊆ 𝑋)
47	46	ralrimiva 3115	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
48		r19.2z 4202	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑀...𝑘) ≠ ∅ ∧ ∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋) → ∃𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
49	33, 47, 48	syl2anr 584	. . . . . . . . . . . . . . . . . 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)
50		iinss 4706	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋 → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ 𝑋)
51	49, 50	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)
52	6	ad2antrr 705	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → 𝐷 ∈ (Met‘𝑋))
53		elfvdm 6363	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ dom Met)
54		fvi 6399	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ dom Met → ( I ‘𝑋) = 𝑋)
55	52, 53, 54	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)
56	51, 55	sseqtr4d 3791	. . . . . . . . . . . . . . . 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))
57		sseqin2 3968	. . . . . . . . . . . . . . . 16 ⊢ (∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ⊆ ( I ‘𝑋) ↔ (( I ‘𝑋) ∩ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)) = ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
58	56, 57	sylib 208	. . . . . . . . . . . . . . 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))
59	30, 58	syl5eqr 2819	. . . . . . . . . . . . . 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))
60	43	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → 𝑔 ∈ (Fil‘𝑋))
61	37	ralrimiva 3115	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) → ∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔)
62	61	adantr 466	. . . . . . . . . . . . . . . . 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)
63	33	adantl 467	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → (𝑀...𝑘) ≠ ∅)
64		fzfid 12979	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ (𝑔 ∈ (CauFil‘𝐷) ∧ 𝑠:ℤ⟶𝑔)) ∧ 𝑘 ∈ 𝑍) → (𝑀...𝑘) ∈ Fin)
65		iinfi 8482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ (∀𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔 ∧ (𝑀...𝑘) ≠ ∅ ∧ (𝑀...𝑘) ∈ Fin)) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ (fi‘𝑔))
66	60, 62, 63, 64, 65	syl13anc 1478	. . . . . . . . . . . . . . . 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))
67		filfi 21882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ (Fil‘𝑋) → (fi‘𝑔) = 𝑔)
68	60, 67	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)
69	66, 68	eleqtrd 2852	. . . . . . . . . . . . . . 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)
70		fileln0 21873	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ (Fil‘𝑋) ∧ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ∈ 𝑔) → ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ≠ ∅)
71	60, 69, 70	syl2anc 573	. . . . . . . . . . . . . 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) ≠ ∅)
72	59, 71	eqnetrd 3010	. . . . . . . . . . . . 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)} ≠ ∅)
73		rabn0 4105	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ ( I ‘𝑋) ∣ 𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)} ≠ ∅ ↔ ∃𝑥 ∈ ( I ‘𝑋)𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛))
74	72, 73	sylib 208	. . . . . . . . . . . 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))
75	74	ralrimiva 3115	. . . . . . . . . . 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))
76	75	adantrrr 704	. . . . . . . . . 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))
77		fvex 6344	. . . . . . . . . . 11 ⊢ ( I ‘𝑋) ∈ V
78		eleq1 2838	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛)))
79		fvex 6344	. . . . . . . . . . . . 13 ⊢ (𝑓‘𝑘) ∈ V
80		eliin 4660	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑘) ∈ V → ((𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))
81	79, 80	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑘) ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛))
82	78, 81	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑘) → (𝑥 ∈ ∩ 𝑛 ∈ (𝑀...𝑘)(𝑠‘𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛)))
83	77, 82	axcc4dom 9468	. . . . . . . . . 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)))
84	29, 76, 83	syl2anc 573	. . . . . . . . 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)))
85		df-ral 3066	. . . . . . . . . . . . 13 ⊢ (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)) ↔ ∀𝑓(𝑓 ∈ (Cau‘𝐷) → (𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))
86		19.29 1953	. . . . . . . . . . . . 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))))
87	85, 86	sylanb 570	. . . . . . . . . . . 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))))
88	24	ad2antrr 705	. . . . . . . . . . . . . . 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)
89	5	ad2antrr 705	. . . . . . . . . . . . . . 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))
90		simprrl 766	. . . . . . . . . . . . . . . 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))
91		feq3 6167	. . . . . . . . . . . . . . . . 17 ⊢ (( I ‘𝑋) = 𝑋 → (𝑓:𝑍⟶( I ‘𝑋) ↔ 𝑓:𝑍⟶𝑋))
92	89, 53, 54, 91	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))
93	90, 92	mpbid 222	. . . . . . . . . . . . . . 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)
94		simplrr 763	. . . . . . . . . . . . . . . . 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)))
95	94	simprd 483	. . . . . . . . . . . . . . . 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))
96		fveq2 6333	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (𝑠‘𝑘) = (𝑠‘𝑖))
97		oveq2 6803	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((1 / 2)↑𝑘) = ((1 / 2)↑𝑖))
98	97	breq2d 4799	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ (𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
99	96, 98	raleqbidv 3301	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
100	96, 99	raleqbidv 3301	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (∀𝑢 ∈ (𝑠‘𝑘)∀𝑣 ∈ (𝑠‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ∀𝑢 ∈ (𝑠‘𝑖)∀𝑣 ∈ (𝑠‘𝑖)(𝑢𝐷𝑣) < ((1 / 2)↑𝑖)))
101	100	cbvralv 3320	. . . . . . . . . . . . . . . 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))
102	95, 101	sylib 208	. . . . . . . . . . . . . . 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))
103		simprrr 767	. . . . . . . . . . . . . . . 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))
104		fveq2 6333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝑠‘𝑛) = (𝑠‘𝑗))
105	104	eleq2d 2836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ (𝑓‘𝑘) ∈ (𝑠‘𝑗)))
106	105	cbvralv 3320	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗))
107		oveq2 6803	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → (𝑀...𝑘) = (𝑀...𝑖))
108		fveq2 6333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → (𝑓‘𝑘) = (𝑓‘𝑖))
109	108	eleq1d 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ (𝑓‘𝑖) ∈ (𝑠‘𝑗)))
110	107, 109	raleqbidv 3301	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (∀𝑗 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑗) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗)))
111	106, 110	syl5bb 272	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗)))
112	111	cbvralv 3320	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝑓‘𝑘) ∈ (𝑠‘𝑛) ↔ ∀𝑖 ∈ 𝑍 ∀𝑗 ∈ (𝑀...𝑖)(𝑓‘𝑖) ∈ (𝑠‘𝑗))
113	103, 112	sylib 208	. . . . . . . . . . . . . . 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))
114	89, 38	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))
115		simplrl 762	. . . . . . . . . . . . . . . 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))
116	114, 115, 42	syl2anc 573	. . . . . . . . . . . . . . 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))
117	94	simpld 482	. . . . . . . . . . . . . . 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)
118	26, 1, 88, 89, 93, 102, 113	iscmet3lem1 23307	. . . . . . . . . . . . . . . 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))
119		simprl 754	. . . . . . . . . . . . . . . 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))))
120	118, 93, 119	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))
121	26, 1, 88, 89, 93, 102, 113, 116, 117, 120	iscmet3lem2 23308	. . . . . . . . . . . . . 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) ≠ ∅)
122	121	ex 397	. . . . . . . . . . . . 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) ≠ ∅))
123	122	exlimdv 2013	. . . . . . . . . . . 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) ≠ ∅))
124	87, 123	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) ≠ ∅))
125	124	expdimp 440	. . . . . . . . . 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) ≠ ∅))
126	125	an32s 631	. . . . . . . . 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) ≠ ∅))
127	84, 126	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) ≠ ∅)
128	127	expr 444	. . . . . . 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) ≠ ∅))
129	128	exlimdv 2013	. . . . . 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) ≠ ∅))
130	23, 129	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) ∧ 𝑔 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑔) ≠ ∅)
131	130	ralrimiva 3115	. . . 4 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅)
132	1	iscmet 23300	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑔 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑔) ≠ ∅))
133	6, 131, 132	sylanbrc 572	. . 3 ⊢ ((𝜑 ∧ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))) → 𝐷 ∈ (CMet‘𝑋))
134	133	ex 397	. 2 ⊢ (𝜑 → (∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)) → 𝐷 ∈ (CMet‘𝑋)))
135	4, 134	impbid2 216	1 ⊢ (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1629   = wceq 1631  ∃wex 1852   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  ∅c0 4063  ∩ ciin 4656   class class class wbr 4787   I cid 5157  dom cdm 5250  ⟶wf 6026  ‘cfv 6030  (class class class)co 6795  ωcom 7215   ≈ cen 8109   ≼ cdom 8110  Fincfn 8112  ficfi 8475  1c1 10142   < clt 10279   / cdiv 10889  ℕcn 11225  2c2 11275  ℤcz 11583  ℤ_≥cuz 11892  ℝ⁺crp 12034  ...cfz 12532  ↑cexp 13066  ∞Metcxmt 19945  Metcme 19946  MetOpencmopn 19950  ⇝_𝑡clm 21250  Filcfil 21868   fLim cflim 21957  CauFilccfil 23268  Caucca 23269  CMetcms 23270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-inf2 8705  ax-cc 9462  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-pre-sup 10219

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-oadd 7720  df-omul 7721  df-er 7899  df-map 8014  df-pm 8015  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-fi 8476  df-sup 8507  df-inf 8508  df-oi 8574  df-card 8968  df-acn 8971  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-div 10890  df-nn 11226  df-2 11284  df-3 11285  df-n0 11499  df-z 11584  df-uz 11893  df-q 11996  df-rp 12035  df-xneg 12150  df-xadd 12151  df-xmul 12152  df-ico 12385  df-fz 12533  df-fl 12800  df-seq 13008  df-exp 13067  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-rlim 14427  df-rest 16290  df-topgen 16311  df-psmet 19952  df-xmet 19953  df-met 19954  df-bl 19955  df-mopn 19956  df-fbas 19957  df-fg 19958  df-top 20918  df-topon 20935  df-bases 20970  df-ntr 21044  df-nei 21122  df-lm 21253  df-fil 21869  df-fm 21961  df-flim 21962  df-flf 21963  df-cfil 23271  df-cau 23272  df-cmet 23273

This theorem is referenced by:  iscmet2  23310  iscmet3i  23328  heibor1  33940  rrncms  33963