Theorem rlimrel 15539

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 15535	. 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 5844	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   class class class wbr 5166  dom cdm 5700  Rel wrel 5705  ‘cfv 6573  (class class class)co 7448   ↑_pm cpm 8885  ℂcc 11182  ℝcr 11183   < clt 11324   ≤ cle 11325   − cmin 11520  ℝ⁺crp 13057  abscabs 15283   ⇝_𝑟 crli 15531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-opab 5229  df-xp 5706  df-rel 5707  df-rlim 15535

This theorem is referenced by:  rlim  15541  rlimpm  15546  rlimdm  15597  caucvgrlem2  15723  caucvgr  15724  rlimdmafv  47092  rlimdmafv2  47173