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Definition df-lm 22725
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 (π‘₯ ∈ ℝ ↦ (sinβ€˜(Ο€ Β· π‘₯))) converges to zero (in the standard topology on the reals) with this definition. (Contributed by NM, 7-Sep-2006.)
Assertion
Ref Expression
df-lm ⇝𝑑 = (𝑗 ∈ Top ↦ {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ∧ π‘₯ ∈ βˆͺ 𝑗 ∧ βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
Distinct variable group:   𝑓,𝑗,π‘₯,𝑦,𝑒

Detailed syntax breakdown of Definition df-lm
StepHypRef Expression
1 clm 22722 . 2 class ⇝𝑑
2 vj . . 3 setvar 𝑗
3 ctop 22387 . . 3 class Top
4 vf . . . . . . 7 setvar 𝑓
54cv 1541 . . . . . 6 class 𝑓
62cv 1541 . . . . . . . 8 class 𝑗
76cuni 4908 . . . . . . 7 class βˆͺ 𝑗
8 cc 11105 . . . . . . 7 class β„‚
9 cpm 8818 . . . . . . 7 class ↑pm
107, 8, 9co 7406 . . . . . 6 class (βˆͺ 𝑗 ↑pm β„‚)
115, 10wcel 2107 . . . . 5 wff 𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚)
12 vx . . . . . . 7 setvar π‘₯
1312cv 1541 . . . . . 6 class π‘₯
1413, 7wcel 2107 . . . . 5 wff π‘₯ ∈ βˆͺ 𝑗
15 vu . . . . . . . 8 setvar 𝑒
1612, 15wel 2108 . . . . . . 7 wff π‘₯ ∈ 𝑒
17 vy . . . . . . . . . 10 setvar 𝑦
1817cv 1541 . . . . . . . . 9 class 𝑦
1915cv 1541 . . . . . . . . 9 class 𝑒
205, 18cres 5678 . . . . . . . . 9 class (𝑓 β†Ύ 𝑦)
2118, 19, 20wf 6537 . . . . . . . 8 wff (𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’
22 cuz 12819 . . . . . . . . 9 class β„€β‰₯
2322crn 5677 . . . . . . . 8 class ran β„€β‰₯
2421, 17, 23wrex 3071 . . . . . . 7 wff βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’
2516, 24wi 4 . . . . . 6 wff (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’)
2625, 15, 6wral 3062 . . . . 5 wff βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’)
2711, 14, 26w3a 1088 . . . 4 wff (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ∧ π‘₯ ∈ βˆͺ 𝑗 ∧ βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))
2827, 4, 12copab 5210 . . 3 class {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ∧ π‘₯ ∈ βˆͺ 𝑗 ∧ βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))}
292, 3, 28cmpt 5231 . 2 class (𝑗 ∈ Top ↦ {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ∧ π‘₯ ∈ βˆͺ 𝑗 ∧ βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
301, 29wceq 1542 1 wff ⇝𝑑 = (𝑗 ∈ Top ↦ {βŸ¨π‘“, π‘₯⟩ ∣ (𝑓 ∈ (βˆͺ 𝑗 ↑pm β„‚) ∧ π‘₯ ∈ βˆͺ 𝑗 ∧ βˆ€π‘’ ∈ 𝑗 (π‘₯ ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝑓 β†Ύ 𝑦):π‘¦βŸΆπ‘’))})
Colors of variables: wff setvar class
This definition is referenced by:  lmrel  22726  lmrcl  22727  lmfval  22728
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