Theorem rlimrel 14842

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 14838	. 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 5658	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  dom cdm 5519  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135   ↑_pm cpm 8390  ℂcc 10524  ℝcr 10525   < clt 10664   ≤ cle 10665   − cmin 10859  ℝ⁺crp 12377  abscabs 14585   ⇝_𝑟 crli 14834

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526  df-rlim 14838

This theorem is referenced by:  rlim  14844  rlimpm  14849  rlimdm  14900  caucvgrlem2  15023  caucvgr  15024  rlimdmafv  43733  rlimdmafv2  43814