Definition df-lm 14858

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 14855	. 2 class ⇝𝑡
2		vj	. . 3 setvar 𝑗
3		ctop 14665	. . 3 class Top
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1394	. . . . . 6 class 𝑓
6	2	cv 1394	. . . . . . . 8 class 𝑗
7	6	cuni 3887	. . . . . . 7 class ∪ 𝑗
8		cc 7993	. . . . . . 7 class ℂ
9		cpm 6794	. . . . . . 7 class ↑pm
10	7, 8, 9	co 6000	. . . . . 6 class (∪ 𝑗 ↑pm ℂ)
11	5, 10	wcel 2200	. . . . 5 wff 𝑓 ∈ (∪ 𝑗 ↑pm ℂ)
12		vx	. . . . . . 7 setvar 𝑥
13	12	cv 1394	. . . . . 6 class 𝑥
14	13, 7	wcel 2200	. . . . 5 wff 𝑥 ∈ ∪ 𝑗
15		vu	. . . . . . . 8 setvar 𝑢
16	12, 15	wel 2201	. . . . . . 7 wff 𝑥 ∈ 𝑢
17		vy	. . . . . . . . . 10 setvar 𝑦
18	17	cv 1394	. . . . . . . . 9 class 𝑦
19	15	cv 1394	. . . . . . . . 9 class 𝑢
20	5, 18	cres 4720	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
21	18, 19, 20	wf 5313	. . . . . . . 8 wff (𝑓 ↾ 𝑦):𝑦⟶𝑢
22		cuz 9718	. . . . . . . . 9 class ℤ≥
23	22	crn 4719	. . . . . . . 8 class ran ℤ≥
24	21, 17, 23	wrex 2509	. . . . . . 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 2508	. . . . 5 wff ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)
27	11, 14, 26	w3a 1002	. . . 4 wff (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))
28	27, 4, 12	copab 4143	. . 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 4144	. 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 1395	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  14859  lmfval  14860