Theorem rlimrel 15202

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 15198	. 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 5730	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   class class class wbr 5074  dom cdm 5589  Rel wrel 5594  ‘cfv 6433  (class class class)co 7275   ↑_pm cpm 8616  ℂcc 10869  ℝcr 10870   < clt 11009   ≤ cle 11010   − cmin 11205  ℝ⁺crp 12730  abscabs 14945   ⇝_𝑟 crli 15194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595  df-rel 5596  df-rlim 15198

This theorem is referenced by:  rlim  15204  rlimpm  15209  rlimdm  15260  caucvgrlem2  15386  caucvgr  15387  rlimdmafv  44669  rlimdmafv2  44750