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Definition df-lm 12348
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 12345 . 2 class 𝑡
2 vj . . 3 setvar 𝑗
3 ctop 12153 . . 3 class Top
4 vf . . . . . . 7 setvar 𝑓
54cv 1330 . . . . . 6 class 𝑓
62cv 1330 . . . . . . . 8 class 𝑗
76cuni 3731 . . . . . . 7 class 𝑗
8 cc 7611 . . . . . . 7 class
9 cpm 6536 . . . . . . 7 class pm
107, 8, 9co 5767 . . . . . 6 class ( 𝑗pm ℂ)
115, 10wcel 1480 . . . . 5 wff 𝑓 ∈ ( 𝑗pm ℂ)
12 vx . . . . . . 7 setvar 𝑥
1312cv 1330 . . . . . 6 class 𝑥
1413, 7wcel 1480 . . . . 5 wff 𝑥 𝑗
15 vu . . . . . . . 8 setvar 𝑢
1612, 15wel 1481 . . . . . . 7 wff 𝑥𝑢
17 vy . . . . . . . . . 10 setvar 𝑦
1817cv 1330 . . . . . . . . 9 class 𝑦
1915cv 1330 . . . . . . . . 9 class 𝑢
205, 18cres 4536 . . . . . . . . 9 class (𝑓𝑦)
2118, 19, 20wf 5114 . . . . . . . 8 wff (𝑓𝑦):𝑦𝑢
22 cuz 9319 . . . . . . . . 9 class
2322crn 4535 . . . . . . . 8 class ran ℤ
2421, 17, 23wrex 2415 . . . . . . 7 wff 𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢
2516, 24wi 4 . . . . . 6 wff (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)
2625, 15, 6wral 2414 . . . . 5 wff 𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢)
2711, 14, 26w3a 962 . . . 4 wff (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))
2827, 4, 12copab 3983 . . 3 class {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))}
292, 3, 28cmpt 3984 . 2 class (𝑗 ∈ Top ↦ {⟨𝑓, 𝑥⟩ ∣ (𝑓 ∈ ( 𝑗pm ℂ) ∧ 𝑥 𝑗 ∧ ∀𝑢𝑗 (𝑥𝑢 → ∃𝑦 ∈ ran ℤ(𝑓𝑦):𝑦𝑢))})
301, 29wceq 1331 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  12349  lmfval  12350
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