Theorem rlimrel 15450

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 15446	. 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 5765	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   class class class wbr 5074  dom cdm 5620  Rel wrel 5625  ‘cfv 6488  (class class class)co 7359   ↑_pm cpm 8768  ℂcc 11032  ℝcr 11033   < clt 11175   ≤ cle 11176   − cmin 11373  ℝ⁺crp 12937  abscabs 15191   ⇝_𝑟 crli 15442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-ss 3901  df-opab 5137  df-xp 5626  df-rel 5627  df-rlim 15446

This theorem is referenced by:  rlim  15452  rlimpm  15457  rlimdm  15508  caucvgrlem2  15632  caucvgr  15633  rlimdmafv  47652  rlimdmafv2  47733