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Theorem rlimrel 14161
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 14157 . 2 𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧𝑤 → (abs‘((𝑓𝑤) − 𝑥)) < 𝑦))}
21relopabi 5207 1 Rel ⇝𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907  wrex 2908   class class class wbr 4615  dom cdm 5076  Rel wrel 5081  cfv 5849  (class class class)co 6607  pm cpm 7806  cc 9881  cr 9882   < clt 10021  cle 10022  cmin 10213  +crp 11779  abscabs 13911  𝑟 crli 14153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-opab 4676  df-xp 5082  df-rel 5083  df-rlim 14157
This theorem is referenced by:  rlim  14163  rlimpm  14168  rlimdm  14219  caucvgrlem2  14342  caucvgr  14343  rlimdmafv  40577
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