Definition df-lm 14369

Description: Define a function on topologies whose value is the convergence relation for sequences into the given topological space. Although 𝑓 is typically a sequence (a function from an upperset of integers) with values in the topological space, it need not be. Note, however, that the limit property concerns only values at integers, so that the real-valued function (𝑥 ∈ ℝ ↦ (sin‘(π · 𝑥))) converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.)

Assertion

Ref	Expression
df-lm	⊢ ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})

Distinct variable group:   𝑓,𝑗,𝑥,𝑦,𝑢

Detailed syntax breakdown of Definition df-lm

Step	Hyp	Ref	Expression
1		clm 14366	. 2 class ⇝𝑡
2		vj	. . 3 setvar 𝑗
3		ctop 14176	. . 3 class Top
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1363	. . . . . 6 class 𝑓
6	2	cv 1363	. . . . . . . 8 class 𝑗
7	6	cuni 3836	. . . . . . 7 class ∪ 𝑗
8		cc 7872	. . . . . . 7 class ℂ
9		cpm 6705	. . . . . . 7 class ↑pm
10	7, 8, 9	co 5919	. . . . . 6 class (∪ 𝑗 ↑pm ℂ)
11	5, 10	wcel 2164	. . . . 5 wff 𝑓 ∈ (∪ 𝑗 ↑pm ℂ)
12		vx	. . . . . . 7 setvar 𝑥
13	12	cv 1363	. . . . . 6 class 𝑥
14	13, 7	wcel 2164	. . . . 5 wff 𝑥 ∈ ∪ 𝑗
15		vu	. . . . . . . 8 setvar 𝑢
16	12, 15	wel 2165	. . . . . . 7 wff 𝑥 ∈ 𝑢
17		vy	. . . . . . . . . 10 setvar 𝑦
18	17	cv 1363	. . . . . . . . 9 class 𝑦
19	15	cv 1363	. . . . . . . . 9 class 𝑢
20	5, 18	cres 4662	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
21	18, 19, 20	wf 5251	. . . . . . . 8 wff (𝑓 ↾ 𝑦):𝑦⟶𝑢
22		cuz 9595	. . . . . . . . 9 class ℤ≥
23	22	crn 4661	. . . . . . . 8 class ran ℤ≥
24	21, 17, 23	wrex 2473	. . . . . . 7 wff ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢
25	16, 24	wi 4	. . . . . 6 wff (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)
26	25, 15, 6	wral 2472	. . . . 5 wff ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)
27	11, 14, 26	w3a 980	. . . 4 wff (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))
28	27, 4, 12	copab 4090	. . 3 class {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}
29	2, 3, 28	cmpt 4091	. 2 class (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})
30	1, 29	wceq 1364	1 wff ⇝𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))})

This definition is referenced by:  lmrcl  14370  lmfval  14371