Theorem climrel 15691

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 15687	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabi 5806	1 ⊢ Rel ⇝

This proof depends on syntax axioms:   ∧ wa 401   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   class class class wbr 5106  Rel wrel 5661  ‘cfv 6561  (class class class)co 7446  ℂcc 11228   < clt 11373   − cmin 11571  ℤcz 12723  ℤ_≥cuz 12995  ℝ⁺crp 13150  abscabs 15433   ⇝ cli 15683

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5172  df-xp 5662  df-rel 5663  df-clim 15687

This theorem is used by:  clim  15693  climcl  15698  climi  15709  climrlim2  15746  fclim  15752  climrecl  15782  climge0  15783  iserex  15856  caurcvg2  15877  caucvg  15878  iseralt  15884  fsumcvg3  15927  cvgcmpce  16017  climfsum  16019  climcnds  16052  trirecip  16064  ntrivcvgn0  16099  ovoliunlem1  25856  mbflimlem  26021  abelthlem5  26795  emcllem6  27361  lgamgulmlem4  27392  binomcxplemnn0  45536  binomcxplemnotnn0  45543  climf  46849  sumnnodd  46857  climf2  46891  climd  46897  clim2d  46898  climfv  46916  climuzlem  46968  climlimsup  46985  climlimsupcex  46994  climliminflimsupd  47026  climliminf  47031  liminflimsupclim  47032  xlimclimdm  47079  ioodvbdlimc1lem2  47157  ioodvbdlimc2lem  47159  stirlinglem12  47310  fouriersw  47456