Theorem rlimrel 14268

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.

Step	Hyp	Ref	Expression
1		df-rlim 14264	. 2 ⊢ ⇝𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))}
2	1	relopabi 5278	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   class class class wbr 4685  dom cdm 5143  Rel wrel 5148  ‘cfv 5926  (class class class)co 6690   ↑_pm cpm 7900  ℂcc 9972  ℝcr 9973   < clt 10112   ≤ cle 10113   − cmin 10304  ℝ⁺crp 11870  abscabs 14018   ⇝_𝑟 crli 14260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150  df-rlim 14264

This theorem is referenced by:  rlim  14270  rlimpm  14275  rlimdm  14326  caucvgrlem2  14449  caucvgr  14450  rlimdmafv  41578