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Theorem rlimrel 14844
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 14840 . 2 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
21relopabi 5689 1 Rel ⇝𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2110  wral 3138  wrex 3139   class class class wbr 5059  dom cdm 5550  Rel wrel 5555  cfv 6350  (class class class)co 7150  pm cpm 8401  cc 10529  cr 10530   < clt 10669  cle 10670  cmin 10864  +crp 12383  abscabs 14587  𝑟 crli 14836
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5122  df-xp 5556  df-rel 5557  df-rlim 14840
This theorem is referenced by:  rlim  14846  rlimpm  14851  rlimdm  14902  caucvgrlem2  15025  caucvgr  15026  rlimdmafv  43369  rlimdmafv2  43450
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