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Theorem rlimrel 14072
 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 14068 . 2 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
21relopabi 5167 1 Rel ⇝𝑟
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  dom cdm 5038  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549   ↑pm cpm 7745  ℂcc 9813  ℝcr 9814   < clt 9953   ≤ cle 9954   − cmin 10145  ℝ+crp 11708  abscabs 13822   ⇝𝑟 crli 14064 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-rlim 14068 This theorem is referenced by:  rlim  14074  rlimpm  14079  rlimdm  14130  caucvgrlem2  14253  caucvgr  14254  rlimdmafv  39906
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