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Theorem climrel 12274
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.
StepHypRef Expression
1 df-clim 12270 . 2  |-  ~~>  =  { <. f ,  y >.  |  ( y  e.  CC  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( ( f `  k )  e.  CC  /\  ( abs `  (
( f `  k
)  -  y ) )  <  x ) ) }
21relopabi 4991 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   Rel wrel 4874   ` cfv 5445  (class class class)co 6072   CCcc 8977    < clt 9109    - cmin 9280   ZZcz 10271   ZZ>=cuz 10477   RR+crp 10601   abscabs 12027    ~~> cli 12266
This theorem is referenced by:  clim  12276  climcl  12281  climi  12292  climrlim2  12329  fclim  12335  climrecl  12365  climge0  12366  iserex  12438  caurcvg2  12459  caucvg  12460  iseralt  12466  fsumcvg3  12511  cvgcmpce  12585  climfsum  12587  climcnds  12619  trirecip  12630  ovoliunlem1  19386  mbflimlem  19547  abelthlem5  20339  emcllem6  20827  lgamgulmlem4  24804  ntrivcvgn0  25215  stirlinglem12  27748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4875  df-rel 4876  df-clim 12270
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