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Theorem climrel 11966
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 11962 . 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 4811 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   CCcc 8735    < clt 8867    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  clim  11968  climcl  11973  climi  11984  climrlim2  12021  fclim  12027  climrecl  12057  climge0  12058  iserex  12130  caurcvg2  12150  caucvg  12151  iseralt  12157  fsumcvg3  12202  cvgcmpce  12276  climfsum  12278  climcnds  12310  trirecip  12321  ovoliunlem1  18861  mbflimlem  19022  abelthlem5  19811  emcllem6  20294  stirlinglem12  27834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-clim 11962
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