Theorem climrel 12048

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 12047	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabi 4905	1 ⊢ Rel ⇝

This proof depends on syntax axioms:   ∧ wa 104   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   class class class wbr 4130  Rel wrel 4779  ‘cfv 5377  (class class class)co 6085  ℂcc 8177   < clt 8360   − cmin 8497  ℤcz 9646  ℤ_≥cuz 9923  ℝ⁺crp 10056  abscabs 11765   ⇝ cli 12046

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-xp 4780  df-rel 4781  df-clim 12047

This theorem is used by:  clim  12049  climcl  12050  climi  12055  fclim  12062  climrecl  12092  iserex  12107  climrecvg1n  12116  climcvg1nlem  12117  fsum3cvg3  12165  trirecip  12270  ntrivcvgap0  12318