Theorem climrel 11799

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 11798	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabi 4847	1 ⊢ Rel ⇝

Syntax hints:   ∧ wa 104   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   class class class wbr 4083  Rel wrel 4724  ‘cfv 5318  (class class class)co 6007  ℂcc 8005   < clt 8189   − cmin 8325  ℤcz 9454  ℤ_≥cuz 9730  ℝ⁺crp 9857  abscabs 11516   ⇝ cli 11797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-xp 4725  df-rel 4726  df-clim 11798

This theorem is referenced by:  clim  11800  climcl  11801  climi  11806  fclim  11813  climrecl  11843  iserex  11858  climrecvg1n  11867  climcvg1nlem  11868  fsum3cvg3  11915  trirecip  12020  ntrivcvgap0  12068