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Theorem climrel 11042
 Description: The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
Assertion
Ref Expression
climrel Rel ⇝

Proof of Theorem climrel
Dummy variables 𝑗 𝑘 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clim 11041 . 2 ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)((𝑓𝑘) ∈ ℂ ∧ (abs‘((𝑓𝑘) − 𝑦)) < 𝑥))}
21relopabi 4660 1 Rel ⇝
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ∈ wcel 1480  ∀wral 2414  ∃wrex 2415   class class class wbr 3924  Rel wrel 4539  ‘cfv 5118  (class class class)co 5767  ℂcc 7611   < clt 7793   − cmin 7926  ℤcz 9047  ℤ≥cuz 9319  ℝ+crp 9434  abscabs 10762   ⇝ cli 11040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-xp 4540  df-rel 4541  df-clim 11041 This theorem is referenced by:  clim  11043  climcl  11044  climi  11049  fclim  11056  climrecl  11086  iserex  11101  climrecvg1n  11110  climcvg1nlem  11111  fsum3cvg3  11158  trirecip  11263  ntrivcvgap0  11311
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