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Definition df-lm 21772
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 21769 . 2 class 𝑡
2 vj . . 3 setvar 𝑗
3 ctop 21436 . . 3 class Top
4 vf . . . . . . 7 setvar 𝑓
54cv 1529 . . . . . 6 class 𝑓
62cv 1529 . . . . . . . 8 class 𝑗
76cuni 4837 . . . . . . 7 class 𝑗
8 cc 10529 . . . . . . 7 class
9 cpm 8402 . . . . . . 7 class pm
107, 8, 9co 7150 . . . . . 6 class ( 𝑗pm ℂ)
115, 10wcel 2107 . . . . 5 wff 𝑓 ∈ ( 𝑗pm ℂ)
12 vx . . . . . . 7 setvar 𝑥
1312cv 1529 . . . . . 6 class 𝑥
1413, 7wcel 2107 . . . . 5 wff 𝑥 𝑗
15 vu . . . . . . . 8 setvar 𝑢
1612, 15wel 2108 . . . . . . 7 wff 𝑥𝑢
17 vy . . . . . . . . . 10 setvar 𝑦
1817cv 1529 . . . . . . . . 9 class 𝑦
1915cv 1529 . . . . . . . . 9 class 𝑢
205, 18cres 5556 . . . . . . . . 9 class (𝑓𝑦)
2118, 19, 20wf 6350 . . . . . . . 8 wff (𝑓𝑦):𝑦𝑢
22 cuz 12237 . . . . . . . . 9 class
2322crn 5555 . . . . . . . 8 class ran ℤ
2421, 17, 23wrex 3144 . . . . . . 7 wff 𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢
2516, 24wi 4 . . . . . 6 wff (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)
2625, 15, 6wral 3143 . . . . 5 wff 𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)
2711, 14, 26w3a 1081 . . . 4 wff (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))
2827, 4, 12copab 5125 . . 3 class {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}
292, 3, 28cmpt 5143 . 2 class (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
301, 29wceq 1530 1 wff 𝑡 = (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
Colors of variables: wff setvar class
This definition is referenced by:  lmrel  21773  lmrcl  21774  lmfval  21775
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