Theorem climrel 14071

Description: The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Assertion

Ref	Expression
climrel	⊢ Rel ⇝

Proof of Theorem climrel

Dummy variables 𝑗 𝑘 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clim 14067	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabi 5167	1 ⊢ Rel ⇝

Syntax hints:   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   < clt 9953   − cmin 10145  ℤcz 11254  ℤ_≥cuz 11563  ℝ⁺crp 11708  abscabs 13822   ⇝ cli 14063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-clim 14067

This theorem is referenced by:  clim  14073  climcl  14078  climi  14089  climrlim2  14126  fclim  14132  climrecl  14162  climge0  14163  iserex  14235  caurcvg2  14256  caucvg  14257  iseralt  14263  fsumcvg3  14307  cvgcmpce  14391  climfsum  14393  climcnds  14422  trirecip  14434  ntrivcvgn0  14469  ovoliunlem1  23077  mbflimlem  23240  abelthlem5  23993  emcllem6  24527  lgamgulmlem4  24558  binomcxplemnn0  37570  binomcxplemnotnn0  37577  climf  38689  sumnnodd  38697  climf2  38733  climd  38739  clim2d  38740  ioodvbdlimc1lem2  38822  ioodvbdlimc2lem  38824  stirlinglem12  38978  fouriersw  39124