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Theorem rlimrel 15530
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 15526 . 2 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
21relopabiv 5829 1 Rel ⇝𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  wral 3060  wrex 3069   class class class wbr 5142  dom cdm 5684  Rel wrel 5689  cfv 6560  (class class class)co 7432  pm cpm 8868  cc 11154  cr 11155   < clt 11296  cle 11297  cmin 11493  +crp 13035  abscabs 15274  𝑟 crli 15522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-opab 5205  df-xp 5690  df-rel 5691  df-rlim 15526
This theorem is referenced by:  rlim  15532  rlimpm  15537  rlimdm  15588  caucvgrlem2  15712  caucvgr  15713  rlimdmafv  47194  rlimdmafv2  47275
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