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Theorem climrel 13274
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 13270 . 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 5126 1  |-  Rel  ~~>
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1767   A.wral 2814   E.wrex 2815   class class class wbr 4447   Rel wrel 5004   ` cfv 5586  (class class class)co 6282   CCcc 9486    < clt 9624    - cmin 9801   ZZcz 10860   ZZ>=cuz 11078   RR+crp 11216   abscabs 13026    ~~> cli 13266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005  df-rel 5006  df-clim 13270
This theorem is referenced by:  clim  13276  climcl  13281  climi  13292  climrlim2  13329  fclim  13335  climrecl  13365  climge0  13366  iserex  13438  caurcvg2  13459  caucvg  13460  iseralt  13466  fsumcvg3  13510  cvgcmpce  13591  climfsum  13593  climcnds  13622  trirecip  13633  ovoliunlem1  21648  mbflimlem  21809  abelthlem5  22564  emcllem6  23058  lgamgulmlem4  28214  ntrivcvgn0  28609  climf  31164  sumnnodd  31172  ioodvbdlimc1lem2  31262  ioodvbdlimc2lem  31264  stirlinglem12  31385  fouriersw  31532
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