Theorem climrel 15502

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 15498	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabiv 5791	1 ⊢ Rel ⇝

Syntax hints:   ∧ wa 399   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   class class class wbr 5099  Rel wrel 5650  ‘cfv 6517  (class class class)co 7392  ℂcc 11068   < clt 11213   − cmin 11411  ℤcz 12565  ℤ_≥cuz 12836  ℝ⁺crp 12990  abscabs 15244   ⇝ cli 15494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3921  df-opab 5162  df-xp 5651  df-rel 5652  df-clim 15498

This theorem is referenced by:  clim  15504  climcl  15509  climi  15520  climrlim2  15557  fclim  15563  climrecl  15593  climge0  15594  iserex  15667  caurcvg2  15688  caucvg  15689  iseralt  15695  fsumcvg3  15739  cvgcmpce  15829  climfsum  15831  climcnds  15864  trirecip  15876  ntrivcvgn0  15911  ovoliunlem1  25544  mbflimlem  25709  abelthlem5  26475  emcllem6  27042  lgamgulmlem4  27073  binomcxplemnn0  44889  binomcxplemnotnn0  44896  climf  46162  sumnnodd  46170  climf2  46204  climd  46210  clim2d  46211  climfv  46229  climuzlem  46281  climlimsup  46298  climlimsupcex  46307  climliminflimsupd  46339  climliminf  46344  liminflimsupclim  46345  xlimclimdm  46392  ioodvbdlimc1lem2  46470  ioodvbdlimc2lem  46472  stirlinglem12  46623  fouriersw  46769