Theorem rlimrel 15437

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 15433	. 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 5821	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   class class class wbr 5149  dom cdm 5677  Rel wrel 5682  ‘cfv 6544  (class class class)co 7409   ↑_pm cpm 8821  ℂcc 11108  ℝcr 11109   < clt 11248   ≤ cle 11249   − cmin 11444  ℝ⁺crp 12974  abscabs 15181   ⇝_𝑟 crli 15429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684  df-rlim 15433

This theorem is referenced by:  rlim  15439  rlimpm  15444  rlimdm  15495  caucvgrlem2  15621  caucvgr  15622  rlimdmafv  45933  rlimdmafv2  46014