Theorem rlimrel 15453

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 15449	. 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 5770	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064   class class class wbr 5079  dom cdm 5625  Rel wrel 5630  ‘cfv 6492  (class class class)co 7363   ↑_pm cpm 8771  ℂcc 11034  ℝcr 11035   < clt 11177   ≤ cle 11178   − cmin 11375  ℝ⁺crp 12940  abscabs 15194   ⇝_𝑟 crli 15445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-ss 3907  df-opab 5142  df-xp 5631  df-rel 5632  df-rlim 15449

This theorem is referenced by:  rlim  15455  rlimpm  15460  rlimdm  15511  caucvgrlem2  15635  caucvgr  15636  rlimdmafv  47647  rlimdmafv2  47728