Theorem rlimrel 15512

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 15508	. 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	relopabiv 5810	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059   class class class wbr 5123  dom cdm 5665  Rel wrel 5670  ‘cfv 6541  (class class class)co 7413   ↑_pm cpm 8849  ℂcc 11135  ℝcr 11136   < clt 11277   ≤ cle 11278   − cmin 11474  ℝ⁺crp 13016  abscabs 15256   ⇝_𝑟 crli 15504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-ss 3948  df-opab 5186  df-xp 5671  df-rel 5672  df-rlim 15508

This theorem is referenced by:  rlim  15514  rlimpm  15519  rlimdm  15570  caucvgrlem2  15694  caucvgr  15695  rlimdmafv  47162  rlimdmafv2  47243