Theorem climrel 14267

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 14263	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabi 5278	1 ⊢ Rel ⇝

Syntax hints:   ∧ wa 383   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   class class class wbr 4685  Rel wrel 5148  ‘cfv 5926  (class class class)co 6690  ℂcc 9972   < clt 10112   − cmin 10304  ℤcz 11415  ℤ_≥cuz 11725  ℝ⁺crp 11870  abscabs 14018   ⇝ cli 14259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150  df-clim 14263

This theorem is referenced by:  clim  14269  climcl  14274  climi  14285  climrlim2  14322  fclim  14328  climrecl  14358  climge0  14359  iserex  14431  caurcvg2  14452  caucvg  14453  iseralt  14459  fsumcvg3  14504  cvgcmpce  14594  climfsum  14596  climcnds  14627  trirecip  14639  ntrivcvgn0  14674  ovoliunlem1  23316  mbflimlem  23479  abelthlem5  24234  emcllem6  24772  lgamgulmlem4  24803  binomcxplemnn0  38865  binomcxplemnotnn0  38872  climf  40172  sumnnodd  40180  climf2  40216  climd  40222  clim2d  40223  climfv  40241  climuzlem  40293  climlimsup  40310  climlimsupcex  40319  climliminflimsupd  40351  climliminf  40356  liminflimsupclim  40357  ioodvbdlimc1lem2  40465  ioodvbdlimc2lem  40467  stirlinglem12  40620  fouriersw  40766