Definition df-lm 23252

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 23249	. 2 class ⇝𝑡
2		vj	. . 3 setvar 𝑗
3		ctop 22914	. . 3 class Top
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1535	. . . . . 6 class 𝑓
6	2	cv 1535	. . . . . . . 8 class 𝑗
7	6	cuni 4911	. . . . . . 7 class ∪ 𝑗
8		cc 11150	. . . . . . 7 class ℂ
9		cpm 8865	. . . . . . 7 class ↑pm
10	7, 8, 9	co 7430	. . . . . 6 class (∪ 𝑗 ↑pm ℂ)
11	5, 10	wcel 2105	. . . . 5 wff 𝑓 ∈ (∪ 𝑗 ↑pm ℂ)
12		vx	. . . . . . 7 setvar 𝑥
13	12	cv 1535	. . . . . 6 class 𝑥
14	13, 7	wcel 2105	. . . . 5 wff 𝑥 ∈ ∪ 𝑗
15		vu	. . . . . . . 8 setvar 𝑢
16	12, 15	wel 2106	. . . . . . 7 wff 𝑥 ∈ 𝑢
17		vy	. . . . . . . . . 10 setvar 𝑦
18	17	cv 1535	. . . . . . . . 9 class 𝑦
19	15	cv 1535	. . . . . . . . 9 class 𝑢
20	5, 18	cres 5690	. . . . . . . . 9 class (𝑓 ↾ 𝑦)
21	18, 19, 20	wf 6558	. . . . . . . 8 wff (𝑓 ↾ 𝑦):𝑦⟶𝑢
22		cuz 12875	. . . . . . . . 9 class ℤ≥
23	22	crn 5689	. . . . . . . 8 class ran ℤ≥
24	21, 17, 23	wrex 3067	. . . . . . 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 3058	. . . . 5 wff ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢)
27	11, 14, 26	w3a 1086	. . . 4 wff (𝑓 ∈ (∪ 𝑗 ↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))
28	27, 4, 12	copab 5209	. . 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 5230	. 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 1536	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  23253  lmrcl  23254  lmfval  23255