Theorem rlimrel 15402

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 15398	. 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 5764	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057   class class class wbr 5093  dom cdm 5619  Rel wrel 5624  ‘cfv 6486  (class class class)co 7352   ↑_pm cpm 8757  ℂcc 11011  ℝcr 11012   < clt 11153   ≤ cle 11154   − cmin 11351  ℝ⁺crp 12892  abscabs 15143   ⇝_𝑟 crli 15394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-ss 3915  df-opab 5156  df-xp 5625  df-rel 5626  df-rlim 15398

This theorem is referenced by:  rlim  15404  rlimpm  15409  rlimdm  15460  caucvgrlem2  15584  caucvgr  15585  rlimdmafv  47301  rlimdmafv2  47382