Theorem climrel 14841

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 14837	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabi 5658	1 ⊢ Rel ⇝

Syntax hints:   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135  ℂcc 10524   < clt 10664   − cmin 10859  ℤcz 11969  ℤ_≥cuz 12231  ℝ⁺crp 12377  abscabs 14585   ⇝ cli 14833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526  df-clim 14837

This theorem is referenced by:  clim  14843  climcl  14848  climi  14859  climrlim2  14896  fclim  14902  climrecl  14932  climge0  14933  iserex  15005  caurcvg2  15026  caucvg  15027  iseralt  15033  fsumcvg3  15078  cvgcmpce  15165  climfsum  15167  climcnds  15198  trirecip  15210  ntrivcvgn0  15246  ovoliunlem1  24106  mbflimlem  24271  abelthlem5  25030  emcllem6  25586  lgamgulmlem4  25617  binomcxplemnn0  41053  binomcxplemnotnn0  41060  climf  42264  sumnnodd  42272  climf2  42308  climd  42314  clim2d  42315  climfv  42333  climuzlem  42385  climlimsup  42402  climlimsupcex  42411  climliminflimsupd  42443  climliminf  42448  liminflimsupclim  42449  xlimclimdm  42496  ioodvbdlimc1lem2  42574  ioodvbdlimc2lem  42576  stirlinglem12  42727  fouriersw  42873