Theorem rlimrel 15503

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 15499	. 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 5791	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   class class class wbr 5099  dom cdm 5645  Rel wrel 5650  ‘cfv 6517  (class class class)co 7392   ↑_pm cpm 8804  ℂcc 11068  ℝcr 11069   < clt 11213   ≤ cle 11214   − cmin 11411  ℝ⁺crp 12990  abscabs 15244   ⇝_𝑟 crli 15495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3921  df-opab 5162  df-xp 5651  df-rel 5652  df-rlim 15499

This theorem is referenced by:  rlim  15505  rlimpm  15510  rlimdm  15561  caucvgrlem2  15685  caucvgr  15686  rlimdmafv  47735  rlimdmafv2  47816