Theorem rlimrel 15534

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 15530	. 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

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   class class class wbr 5105  dom cdm 5652  Rel wrel 5657  ‘cfv 6525  (class class class)co 7400   ↑_pm cpm 8813  ℂcc 11086  ℝcr 11087   < clt 11231   ≤ cle 11232   − cmin 11429  ℝ⁺crp 13007  abscabs 15275   ⇝_𝑟 crli 15526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3924  df-opab 5168  df-xp 5658  df-rel 5659  df-rlim 15530

This theorem is referenced by:  rlim  15536  rlimpm  15541  rlimdm  15592  caucvgrlem2  15716  caucvgr  15717  rlimdmafv  47769  rlimdmafv2  47850