Theorem rlimrel 15514

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 15510	. 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 5804	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   class class class wbr 5124  dom cdm 5659  Rel wrel 5664  ‘cfv 6536  (class class class)co 7410   ↑_pm cpm 8846  ℂcc 11132  ℝcr 11133   < clt 11274   ≤ cle 11275   − cmin 11471  ℝ⁺crp 13013  abscabs 15258   ⇝_𝑟 crli 15506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-ss 3948  df-opab 5187  df-xp 5665  df-rel 5666  df-rlim 15510

This theorem is referenced by:  rlim  15516  rlimpm  15521  rlimdm  15572  caucvgrlem2  15696  caucvgr  15697  rlimdmafv  47173  rlimdmafv2  47254