Definition df-lm 14920

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 14917	. 2 class ⇝𝑡
2		vj	. . 3 setvar 𝑗
3		ctop 14727	. . 3 class Top
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1396	. . . . . 6 class 𝑓
6	2	cv 1396	. . . . . . . 8 class 𝑗
7	6	cuni 3893	. . . . . . 7 class ∪ 𝑗
8		cc 8030	. . . . . . 7 class ℂ
9		cpm 6818	. . . . . . 7 class ↑pm
10	7, 8, 9	co 6018	. . . . . 6 class (∪ 𝑗 ↑pm ℂ)
11	5, 10	wcel 2202	. . . . 5 wff 𝑓 ∈ (∪ 𝑗 ↑pm ℂ)
12		vx	. . . . . . 7 setvar 𝑥
13	12	cv 1396	. . . . . 6 class 𝑥
14	13, 7	wcel 2202	. . . . 5 wff 𝑥 ∈ ∪ 𝑗
15		vu	. . . . . . . 8 setvar 𝑢
16	12, 15	wel 2203	. . . . . . 7 wff 𝑥 ∈ 𝑢
17		vy	. . . . . . . . . 10 setvar 𝑦
18	17	cv 1396	. . . . . . . . 9 class 𝑦
19	15	cv 1396	. . . . . . . . 9 class 𝑢
20	5, 18	cres 4727	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
21	18, 19, 20	wf 5322	. . . . . . . 8 wff (𝑓 ↾ 𝑦):𝑦⟶𝑢
22		cuz 9755	. . . . . . . . 9 class ℤ≥
23	22	crn 4726	. . . . . . . 8 class ran ℤ≥
24	21, 17, 23	wrex 2511	. . . . . . 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 2510	. . . . 5 wff ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)
27	11, 14, 26	w3a 1004	. . . 4 wff (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))
28	27, 4, 12	copab 4149	. . 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 4150	. 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 1397	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:  lmrel  14921  lmrcl  14922  lmfval  14923