Theorem rlimrel 15463

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 15459	. 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 5816	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2099  ∀wral 3057  ∃wrex 3066   class class class wbr 5142  dom cdm 5672  Rel wrel 5677  ‘cfv 6542  (class class class)co 7414   ↑_pm cpm 8839  ℂcc 11130  ℝcr 11131   < clt 11272   ≤ cle 11273   − cmin 11468  ℝ⁺crp 13000  abscabs 15207   ⇝_𝑟 crli 15455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-in 3952  df-ss 3962  df-opab 5205  df-xp 5678  df-rel 5679  df-rlim 15459

This theorem is referenced by:  rlim  15465  rlimpm  15470  rlimdm  15521  caucvgrlem2  15647  caucvgr  15648  rlimdmafv  46551  rlimdmafv2  46632