Theorem rlimrel 15441

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 15437	. 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 5820	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   class class class wbr 5148  dom cdm 5676  Rel wrel 5681  ‘cfv 6543  (class class class)co 7411   ↑_pm cpm 8823  ℂcc 11110  ℝcr 11111   < clt 11252   ≤ cle 11253   − cmin 11448  ℝ⁺crp 12978  abscabs 15185   ⇝_𝑟 crli 15433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-rel 5683  df-rlim 15437

This theorem is referenced by:  rlim  15443  rlimpm  15448  rlimdm  15499  caucvgrlem2  15625  caucvgr  15626  rlimdmafv  46184  rlimdmafv2  46265