Theorem rlimrel 14850

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 14846	. 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 5694	1 ⊢ Rel ⇝𝑟

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   class class class wbr 5066  dom cdm 5555  Rel wrel 5560  ‘cfv 6355  (class class class)co 7156   ↑_pm cpm 8407  ℂcc 10535  ℝcr 10536   < clt 10675   ≤ cle 10676   − cmin 10870  ℝ⁺crp 12390  abscabs 14593   ⇝_𝑟 crli 14842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5129  df-xp 5561  df-rel 5562  df-rlim 14846

This theorem is referenced by:  rlim  14852  rlimpm  14857  rlimdm  14908  caucvgrlem2  15031  caucvgr  15032  rlimdmafv  43396  rlimdmafv2  43477