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Theorem rlimrel 15526
Description: The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
Assertion
Ref Expression
rlimrel Rel ⇝𝑟

Proof of Theorem rlimrel
Dummy variables 𝑤 𝑥 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rlim 15522 . 2 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
21relopabiv 5833 1 Rel ⇝𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059  wrex 3068   class class class wbr 5148  dom cdm 5689  Rel wrel 5694  cfv 6563  (class class class)co 7431  pm cpm 8866  cc 11151  cr 11152   < clt 11293  cle 11294  cmin 11490  +crp 13032  abscabs 15270  𝑟 crli 15518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695  df-rel 5696  df-rlim 15522
This theorem is referenced by:  rlim  15528  rlimpm  15533  rlimdm  15584  caucvgrlem2  15708  caucvgr  15709  rlimdmafv  47127  rlimdmafv2  47208
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