Theorem climrel 11757

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 11756	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabi 4824	1 ⊢ Rel ⇝

Syntax hints:   ∧ wa 104   ∈ wcel 2180  ∀wral 2488  ∃wrex 2489   class class class wbr 4062  Rel wrel 4701  ‘cfv 5294  (class class class)co 5974  ℂcc 7965   < clt 8149   − cmin 8285  ℤcz 9414  ℤ_≥cuz 9690  ℝ⁺crp 9817  abscabs 11474   ⇝ cli 11755

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-opab 4125  df-xp 4702  df-rel 4703  df-clim 11756

This theorem is referenced by:  clim  11758  climcl  11759  climi  11764  fclim  11771  climrecl  11801  iserex  11816  climrecvg1n  11825  climcvg1nlem  11826  fsum3cvg3  11873  trirecip  11978  ntrivcvgap0  12026