Theorem rlimrel 15640

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 15636	. 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 5798	1 ⊢ Rel ⇝𝑟

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   class class class wbr 5103  dom cdm 5651  Rel wrel 5656  ‘cfv 6531  (class class class)co 7412   ↑_pm cpm 8832  ℂcc 11179  ℝcr 11180   < clt 11324   ≤ cle 11325   − cmin 11522  ℝ⁺crp 13101  abscabs 15381   ⇝_𝑟 crli 15632

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-ss 3916  df-opab 5168  df-xp 5657  df-rel 5658  df-rlim 15636

This theorem is used by:  rlim  15642  rlimpm  15647  rlimdm  15698  caucvgrlem2  15822  caucvgr  15823  rlimdmafv  48191  rlimdmafv2  48272