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Definition df-lm 12590
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 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
Distinct variable group:   𝑓,𝑗,𝑥,𝑦,𝑢

Detailed syntax breakdown of Definition df-lm
StepHypRef Expression
1 clm 12587 . 2 class 𝑡
2 vj . . 3 setvar 𝑗
3 ctop 12395 . . 3 class Top
4 vf . . . . . . 7 setvar 𝑓
54cv 1334 . . . . . 6 class 𝑓
62cv 1334 . . . . . . . 8 class 𝑗
76cuni 3772 . . . . . . 7 class 𝑗
8 cc 7730 . . . . . . 7 class
9 cpm 6594 . . . . . . 7 class pm
107, 8, 9co 5824 . . . . . 6 class ( 𝑗pm ℂ)
115, 10wcel 2128 . . . . 5 wff 𝑓 ∈ ( 𝑗pm ℂ)
12 vx . . . . . . 7 setvar 𝑥
1312cv 1334 . . . . . 6 class 𝑥
1413, 7wcel 2128 . . . . 5 wff 𝑥 𝑗
15 vu . . . . . . . 8 setvar 𝑢
1612, 15wel 2129 . . . . . . 7 wff 𝑥𝑢
17 vy . . . . . . . . . 10 setvar 𝑦
1817cv 1334 . . . . . . . . 9 class 𝑦
1915cv 1334 . . . . . . . . 9 class 𝑢
205, 18cres 4588 . . . . . . . . 9 class (𝑓𝑦)
2118, 19, 20wf 5166 . . . . . . . 8 wff (𝑓𝑦):𝑦𝑢
22 cuz 9439 . . . . . . . . 9 class
2322crn 4587 . . . . . . . 8 class ran ℤ
2421, 17, 23wrex 2436 . . . . . . 7 wff 𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢
2516, 24wi 4 . . . . . 6 wff (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)
2625, 15, 6wral 2435 . . . . 5 wff 𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)
2711, 14, 26w3a 963 . . . 4 wff (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))
2827, 4, 12copab 4024 . . 3 class {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}
292, 3, 28cmpt 4025 . 2 class (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
301, 29wceq 1335 1 wff 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
Colors of variables: wff set class
This definition is referenced by:  lmrcl  12591  lmfval  12592
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