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Theorem rlimrel 14838
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 14834 . 2 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
21relopabi 5687 1 Rel ⇝𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wral 3135  wrex 3136   class class class wbr 5057  dom cdm 5548  Rel wrel 5553  cfv 6348  (class class class)co 7145  pm cpm 8396  cc 10523  cr 10524   < clt 10663  cle 10664  cmin 10858  +crp 12377  abscabs 14581  𝑟 crli 14830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554  df-rel 5555  df-rlim 14834
This theorem is referenced by:  rlim  14840  rlimpm  14845  rlimdm  14896  caucvgrlem2  15019  caucvgr  15020  rlimdmafv  43253  rlimdmafv2  43334
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