Theorem climrel 15465

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 15461	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabiv 5786	1 ⊢ Rel ⇝

Syntax hints:   ∧ wa 395   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   class class class wbr 5110  Rel wrel 5646  ‘cfv 6514  (class class class)co 7390  ℂcc 11073   < clt 11215   − cmin 11412  ℤcz 12536  ℤ_≥cuz 12800  ℝ⁺crp 12958  abscabs 15207   ⇝ cli 15457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3934  df-opab 5173  df-xp 5647  df-rel 5648  df-clim 15461

This theorem is referenced by:  clim  15467  climcl  15472  climi  15483  climrlim2  15520  fclim  15526  climrecl  15556  climge0  15557  iserex  15630  caurcvg2  15651  caucvg  15652  iseralt  15658  fsumcvg3  15702  cvgcmpce  15791  climfsum  15793  climcnds  15824  trirecip  15836  ntrivcvgn0  15871  ovoliunlem1  25410  mbflimlem  25575  abelthlem5  26352  emcllem6  26918  lgamgulmlem4  26949  binomcxplemnn0  44345  binomcxplemnotnn0  44352  climf  45627  sumnnodd  45635  climf2  45671  climd  45677  clim2d  45678  climfv  45696  climuzlem  45748  climlimsup  45765  climlimsupcex  45774  climliminflimsupd  45806  climliminf  45811  liminflimsupclim  45812  xlimclimdm  45859  ioodvbdlimc1lem2  45937  ioodvbdlimc2lem  45939  stirlinglem12  46090  fouriersw  46236