Theorem climrel 11958

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 11957	. 2 ⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
2	1	relopabi 4879	1 ⊢ Rel ⇝

Syntax hints:   ∧ wa 104   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521   class class class wbr 4108  Rel wrel 4753  ‘cfv 5351  (class class class)co 6049  ℂcc 8121   < clt 8304   − cmin 8440  ℤcz 9573  ℤ_≥cuz 9849  ℝ⁺crp 9982  abscabs 11675   ⇝ cli 11956

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-opab 4171  df-xp 4754  df-rel 4755  df-clim 11957

This theorem is referenced by:  clim  11959  climcl  11960  climi  11965  fclim  11972  climrecl  12002  iserex  12017  climrecvg1n  12026  climcvg1nlem  12027  fsum3cvg3  12075  trirecip  12180  ntrivcvgap0  12228