Theorem rlimrel 15570

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 15566	. 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 5812	1 ⊢ Rel ⇝𝑟

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092   class class class wbr 5114  dom cdm 5666  Rel wrel 5671  ‘cfv 6543  (class class class)co 7423   ↑_pm cpm 8834  ℂcc 11116  ℝcr 11117   < clt 11261   ≤ cle 11262   − cmin 11459  ℝ⁺crp 13034  abscabs 15311   ⇝_𝑟 crli 15562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-ss 3925  df-opab 5179  df-xp 5672  df-rel 5673  df-rlim 15566

This theorem is used by:  rlim  15572  rlimpm  15577  rlimdm  15628  caucvgrlem2  15752  caucvgr  15753  rlimdmafv  47955  rlimdmafv2  48036