Theorem rlimrel 15530

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 15526	. 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 5829	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069   class class class wbr 5142  dom cdm 5684  Rel wrel 5689  ‘cfv 6560  (class class class)co 7432   ↑_pm cpm 8868  ℂcc 11154  ℝcr 11155   < clt 11296   ≤ cle 11297   − cmin 11493  ℝ⁺crp 13035  abscabs 15274   ⇝_𝑟 crli 15522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-opab 5205  df-xp 5690  df-rel 5691  df-rlim 15526

This theorem is referenced by:  rlim  15532  rlimpm  15537  rlimdm  15588  caucvgrlem2  15712  caucvgr  15713  rlimdmafv  47194  rlimdmafv2  47275