Definition df-lm 23438

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 23435	. 2 class ⇝𝑡
2		vj	. . 3 setvar 𝑗
3		ctop 23102	. . 3 class Top
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1569	. . . . . 6 class 𝑓
6	2	cv 1569	. . . . . . . 8 class 𝑗
7	6	cuni 4874	. . . . . . 7 class ∪ 𝑗
8		cc 11115	. . . . . . 7 class ℂ
9		cpm 8831	. . . . . . 7 class ↑pm
10	7, 8, 9	co 7419	. . . . . 6 class (∪ 𝑗 ↑pm ℂ)
11	5, 10	wcel 2146	. . . . 5 wff 𝑓 ∈ (∪ 𝑗 ↑pm ℂ)
12		vx	. . . . . . 7 setvar 𝑥
13	12	cv 1569	. . . . . 6 class 𝑥
14	13, 7	wcel 2146	. . . . 5 wff 𝑥 ∈ ∪ 𝑗
15		vu	. . . . . . . 8 setvar 𝑢
16	12, 15	wel 2147	. . . . . . 7 wff 𝑥 ∈ 𝑢
17		vy	. . . . . . . . . 10 setvar 𝑦
18	17	cv 1569	. . . . . . . . 9 class 𝑦
19	15	cv 1569	. . . . . . . . 9 class 𝑢
20	5, 18	cres 5665	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
21	18, 19, 20	wf 6536	. . . . . . . 8 wff (𝑓 ↾ 𝑦):𝑦⟶𝑢
22		cuz 12880	. . . . . . . . 9 class ℤ≥
23	22	crn 5664	. . . . . . . 8 class ran ℤ≥
24	21, 17, 23	wrex 3091	. . . . . . 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 3081	. . . . 5 wff ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)
27	11, 14, 26	w3a 1103	. . . 4 wff (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))
28	27, 4, 12	copab 5175	. . 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 5194	. 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 1570	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 used by:  lmrel  23439  lmrcl  23440  lmfval  23441