Theorem rlimrel 14600

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 14596	. 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	relopabi 5477	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2166  ∀wral 3116  ∃wrex 3117   class class class wbr 4872  dom cdm 5341  Rel wrel 5346  ‘cfv 6122  (class class class)co 6904   ↑_pm cpm 8122  ℂcc 10249  ℝcr 10250   < clt 10390   ≤ cle 10391   − cmin 10584  ℝ⁺crp 12111  abscabs 14350   ⇝_𝑟 crli 14592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-opab 4935  df-xp 5347  df-rel 5348  df-rlim 14596

This theorem is referenced by:  rlim  14602  rlimpm  14607  rlimdm  14658  caucvgrlem2  14781  caucvgr  14782  rlimdmafv  42078  rlimdmafv2  42159