Theorem rlimrel 15428

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 15424	. 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 5777	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   class class class wbr 5100  dom cdm 5632  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368   ↑_pm cpm 8776  ℂcc 11036  ℝcr 11037   < clt 11178   ≤ cle 11179   − cmin 11376  ℝ⁺crp 12917  abscabs 15169   ⇝_𝑟 crli 15420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-opab 5163  df-xp 5638  df-rel 5639  df-rlim 15424

This theorem is referenced by:  rlim  15430  rlimpm  15435  rlimdm  15486  caucvgrlem2  15610  caucvgr  15611  rlimdmafv  47531  rlimdmafv2  47612