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Theorem climrel 12062
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 12058 . 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 4893 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1710   A.wral 2619   E.wrex 2620   class class class wbr 4104   Rel wrel 4776   ` cfv 5337  (class class class)co 5945   CCcc 8825    < clt 8957    - cmin 9127   ZZcz 10116   ZZ>=cuz 10322   RR+crp 10446   abscabs 11815    ~~> cli 12054
This theorem is referenced by:  clim  12064  climcl  12069  climi  12080  climrlim2  12117  fclim  12123  climrecl  12153  climge0  12154  iserex  12226  caurcvg2  12247  caucvg  12248  iseralt  12254  fsumcvg3  12299  cvgcmpce  12373  climfsum  12375  climcnds  12407  trirecip  12418  ovoliunlem1  18965  mbflimlem  19126  abelthlem5  19918  emcllem6  20406  lgamgulmlem4  24065  ntrivcvgn0  24527  stirlinglem12  27157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-opab 4159  df-xp 4777  df-rel 4778  df-clim 12058
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