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Theorem caucfil 24791

Description: A Cauchy sequence predicate can be expressed in terms of the Cauchy filter predicate for a suitably chosen filter. (Contributed by Mario Carneiro, 13-Oct-2015.)

**Hypotheses**
Ref	Expression
caucfil.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
caucfil.2	⊢ 𝐿 = ((𝑋 FilMap 𝐹)‘(ℤ_≥ “ 𝑍))

**Assertion**
Ref	Expression
caucfil	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐿 ∈ (CauFil‘𝐷)))

Proof of Theorem caucfil Dummy variables 𝑗 𝑘 𝑚 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 1089	. . . . . . . 8 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
2		caucfil.1	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3	2	uztrn2 12837	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
4	3	adantll 712	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
5		simpll3 1214	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶𝑋)
6	5	fdmd 6725	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
7	4, 6	eleqtrrd 2836	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
8	5, 4	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ 𝑋)
9	7, 8	jca 512	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋))
10	9	biantrurd 533	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
11		uzss 12841	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑗))
12	11	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑗))
13	12	sseld 3980	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑚 ∈ (ℤ_≥‘𝑘) → 𝑚 ∈ (ℤ_≥‘𝑗)))
14	13	pm4.71rd 563	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑚 ∈ (ℤ_≥‘𝑘) ↔ (𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑘))))
15	14	imbi1d 341	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
16		impexp 451	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ (𝑚 ∈ (ℤ_≥‘𝑗) → (𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
17	15, 16	bitrdi 286	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ (𝑚 ∈ (ℤ_≥‘𝑗) → (𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))
18	17	ralbidv2 3173	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
19	10, 18	bitr3d 280	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
20	1, 19	bitrid 282	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
21	20	ralbidva 3175	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
22		r19.26-2 3138	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥)) ↔ (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥)))
23		eleq1w 2816	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑘 → (𝑢 ∈ (ℤ_≥‘𝑚) ↔ 𝑘 ∈ (ℤ_≥‘𝑚)))
24		fveq2 6888	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑘 → (𝐹‘𝑢) = (𝐹‘𝑘))
25	24	oveq2d 7421	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑘 → ((𝐹‘𝑚)𝐷(𝐹‘𝑢)) = ((𝐹‘𝑚)𝐷(𝐹‘𝑘)))
26	25	breq1d 5157	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑘 → (((𝐹‘𝑚)𝐷(𝐹‘𝑢)) < 𝑥 ↔ ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥))
27	23, 26	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑘 → ((𝑢 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑢)) < 𝑥) ↔ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥)))
28	27	cbvralvw 3234	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ (ℤ_≥‘𝑗)(𝑢 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑢)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥))
29	28	ralbii 3093	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑢 ∈ (ℤ_≥‘𝑗)(𝑢 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑢)) < 𝑥) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥))
30		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (ℤ_≥‘𝑚) = (ℤ_≥‘𝑘))
31	30	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (𝑢 ∈ (ℤ_≥‘𝑚) ↔ 𝑢 ∈ (ℤ_≥‘𝑘)))
32		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
33	32	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚)𝐷(𝐹‘𝑢)) = ((𝐹‘𝑘)𝐷(𝐹‘𝑢)))
34	33	breq1d 5157	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (((𝐹‘𝑚)𝐷(𝐹‘𝑢)) < 𝑥 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑢)) < 𝑥))
35	31, 34	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((𝑢 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑢)) < 𝑥) ↔ (𝑢 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑢)) < 𝑥)))
36		eleq1w 2816	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (𝑢 ∈ (ℤ_≥‘𝑘) ↔ 𝑚 ∈ (ℤ_≥‘𝑘)))
37		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑚 → (𝐹‘𝑢) = (𝐹‘𝑚))
38	37	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑚 → ((𝐹‘𝑘)𝐷(𝐹‘𝑢)) = ((𝐹‘𝑘)𝐷(𝐹‘𝑚)))
39	38	breq1d 5157	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑚 → (((𝐹‘𝑘)𝐷(𝐹‘𝑢)) < 𝑥 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
40	36, 39	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑚 → ((𝑢 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑢)) < 𝑥) ↔ (𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
41	35, 40	cbvral2vw 3238	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑢 ∈ (ℤ_≥‘𝑗)(𝑢 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑢)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
42		ralcom 3286	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥))
43	29, 41, 42	3bitr3i 300	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥))
44	43	anbi2i 623	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)) ↔ (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥)))
45		anidm 565	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
46	22, 44, 45	3bitr2i 298	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥)) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
47		simpll1 1212	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → 𝐷 ∈ (∞Met‘𝑋))
48		simpll3 1214	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → 𝐹:𝑍⟶𝑋)
49	2	uztrn2 12837	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → 𝑚 ∈ 𝑍)
50	49	ad2ant2l 744	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → 𝑚 ∈ 𝑍)
51	48, 50	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑚) ∈ 𝑋)
52	8	adantrr 715	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ 𝑋)
53		xmetsym 23844	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑚) ∈ 𝑋 ∧ (𝐹‘𝑘) ∈ 𝑋) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(𝐹‘𝑚)))
54	47, 51, 52, 53	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(𝐹‘𝑚)))
55	54	breq1d 5157	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
56	55	imbi2d 340	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → ((𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥) ↔ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
57	56	anbi2d 629	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥)) ↔ ((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))
58		jaob 960	. . . . . . . . . 10 ⊢ (((𝑚 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑚)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
59		eluzelz 12828	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → 𝑘 ∈ ℤ)
60		eluzelz 12828	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ℤ_≥‘𝑗) → 𝑚 ∈ ℤ)
61		uztric 12842	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑚)))
62	59, 60, 61	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗)) → (𝑚 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑚)))
63	62	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (𝑚 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑚)))
64		pm5.5 361	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑚)) → (((𝑚 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑚)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
65	63, 64	syl 17	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (((𝑚 ∈ (ℤ_≥‘𝑘) ∨ 𝑘 ∈ (ℤ_≥‘𝑚)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
66	58, 65	bitr3id 284	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)) ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
67	57, 66	bitrd 278	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) ∧ (𝑘 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑗))) → (((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥)) ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
68	67	2ralbidva 3216	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ∧ (𝑘 ∈ (ℤ_≥‘𝑚) → ((𝐹‘𝑚)𝐷(𝐹‘𝑘)) < 𝑥)) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
69	46, 68	bitr3id 284	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)(𝑚 ∈ (ℤ_≥‘𝑘) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
70	21, 69	bitrd 278	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
71	70	rexbidva 3176	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
72		uzf 12821	. . . . . 6 ⊢ ℤ_≥:ℤ⟶𝒫 ℤ
73		ffn 6714	. . . . . 6 ⊢ (ℤ_≥:ℤ⟶𝒫 ℤ → ℤ_≥ Fn ℤ)
74	72, 73	ax-mp 5	. . . . 5 ⊢ ℤ_≥ Fn ℤ
75		uzssz 12839	. . . . . 6 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
76	2, 75	eqsstri 4015	. . . . 5 ⊢ 𝑍 ⊆ ℤ
77		raleq 3322	. . . . . . 7 ⊢ (𝑢 = (ℤ_≥‘𝑗) → (∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
78	77	raleqbi1dv 3333	. . . . . 6 ⊢ (𝑢 = (ℤ_≥‘𝑗) → (∀𝑘 ∈ 𝑢 ∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
79	78	rexima 7235	. . . . 5 ⊢ ((ℤ_≥ Fn ℤ ∧ 𝑍 ⊆ ℤ) → (∃𝑢 ∈ (ℤ_≥ “ 𝑍)∀𝑘 ∈ 𝑢 ∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
80	74, 76, 79	mp2an 690	. . . 4 ⊢ (∃𝑢 ∈ (ℤ_≥ “ 𝑍)∀𝑘 ∈ 𝑢 ∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)
81	71, 80	bitr4di 288	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∃𝑢 ∈ (ℤ_≥ “ 𝑍)∀𝑘 ∈ 𝑢 ∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
82	81	ralbidv 3177	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑢 ∈ (ℤ_≥ “ 𝑍)∀𝑘 ∈ 𝑢 ∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
83		elfvdm 6925	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
84	83	adantr 481	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ) → 𝑋 ∈ dom ∞Met)
85		cnex 11187	. . . . . 6 ⊢ ℂ ∈ V
86	84, 85	jctir 521	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ) → (𝑋 ∈ dom ∞Met ∧ ℂ ∈ V))
87		zsscn 12562	. . . . . . 7 ⊢ ℤ ⊆ ℂ
88	76, 87	sstri 3990	. . . . . 6 ⊢ 𝑍 ⊆ ℂ
89	88	jctr 525	. . . . 5 ⊢ (𝐹:𝑍⟶𝑋 → (𝐹:𝑍⟶𝑋 ∧ 𝑍 ⊆ ℂ))
90		elpm2r 8835	. . . . 5 ⊢ (((𝑋 ∈ dom ∞Met ∧ ℂ ∈ V) ∧ (𝐹:𝑍⟶𝑋 ∧ 𝑍 ⊆ ℂ)) → 𝐹 ∈ (𝑋 ↑_pm ℂ))
91	86, 89, 90	syl2an 596	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ) ∧ 𝐹:𝑍⟶𝑋) → 𝐹 ∈ (𝑋 ↑_pm ℂ))
92		simpl 483	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ) → 𝐷 ∈ (∞Met‘𝑋))
93		simpr 485	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
94	2, 92, 93	iscau3 24786	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))
95	94	baibd 540	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ) ∧ 𝐹 ∈ (𝑋 ↑_pm ℂ)) → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
96	91, 95	syldan 591	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ) ∧ 𝐹:𝑍⟶𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
97	96	3impa 1110	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
98		caucfil.2	. . . 4 ⊢ 𝐿 = ((𝑋 FilMap 𝐹)‘(ℤ_≥ “ 𝑍))
99	98	eleq1i 2824	. . 3 ⊢ (𝐿 ∈ (CauFil‘𝐷) ↔ ((𝑋 FilMap 𝐹)‘(ℤ_≥ “ 𝑍)) ∈ (CauFil‘𝐷))
100	2	uzfbas 23393	. . . 4 ⊢ (𝑀 ∈ ℤ → (ℤ_≥ “ 𝑍) ∈ (fBas‘𝑍))
101		fmcfil 24780	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (ℤ_≥ “ 𝑍) ∈ (fBas‘𝑍) ∧ 𝐹:𝑍⟶𝑋) → (((𝑋 FilMap 𝐹)‘(ℤ_≥ “ 𝑍)) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑢 ∈ (ℤ_≥ “ 𝑍)∀𝑘 ∈ 𝑢 ∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
102	100, 101	syl3an2 1164	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (((𝑋 FilMap 𝐹)‘(ℤ_≥ “ 𝑍)) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑢 ∈ (ℤ_≥ “ 𝑍)∀𝑘 ∈ 𝑢 ∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
103	99, 102	bitrid 282	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐿 ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑢 ∈ (ℤ_≥ “ 𝑍)∀𝑘 ∈ 𝑢 ∀𝑚 ∈ 𝑢 ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
104	82, 97, 103	3bitr4d 310	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐿 ∈ (CauFil‘𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⊆ wss 3947 𝒫 cpw 4601 class class class wbr 5147 dom cdm 5675 “ cima 5678 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_pm cpm 8817 ℂcc 11104 < clt 11244 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ∞Metcxmet 20921 fBascfbas 20924 FilMap cfm 23428 CauFilccfil 24760 Cauccau 24761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-z 12555 df-uz 12819 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-rest 17364 df-psmet 20928 df-xmet 20929 df-bl 20931 df-fbas 20933 df-fg 20934 df-fil 23341 df-fm 23433 df-cfil 24763 df-cau 24764

This theorem is referenced by: cmetcaulem 24796

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