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Theorem climrel 11288
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 11287 . 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 4753 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 104    e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4004   Rel wrel 4632   ` cfv 5217  (class class class)co 5875   CCcc 7809    < clt 7992    - cmin 8128   ZZcz 9253   ZZ>=cuz 9528   RR+crp 9653   abscabs 11006    ~~> cli 11286
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-opab 4066  df-xp 4633  df-rel 4634  df-clim 11287
This theorem is referenced by:  clim  11289  climcl  11290  climi  11295  fclim  11302  climrecl  11332  iserex  11347  climrecvg1n  11356  climcvg1nlem  11357  fsum3cvg3  11404  trirecip  11509  ntrivcvgap0  11557
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