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Theorem rlimrel 15463
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 15459 . 2 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
21relopabiv 5816 1 Rel ⇝𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  wral 3057  wrex 3066   class class class wbr 5142  dom cdm 5672  Rel wrel 5677  cfv 6542  (class class class)co 7414  pm cpm 8839  cc 11130  cr 11131   < clt 11272  cle 11273  cmin 11468  +crp 13000  abscabs 15207  𝑟 crli 15455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-in 3952  df-ss 3962  df-opab 5205  df-xp 5678  df-rel 5679  df-rlim 15459
This theorem is referenced by:  rlim  15465  rlimpm  15470  rlimdm  15521  caucvgrlem2  15647  caucvgr  15648  rlimdmafv  46551  rlimdmafv2  46632
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