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Theorem climrel 11961
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  ~~>
Dummy variables  j 
k  x  y  f are mutually distinct and distinct from all other variables.

Proof of Theorem climrel
StepHypRef Expression
1 df-clim 11957 . 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 360    e. wcel 1685   A.wral 2545   E.wrex 2546   class class class wbr 4025   Rel wrel 4694   ` cfv 5222  (class class class)co 5820   CCcc 8731    < clt 8863    - cmin 9033   ZZcz 10020   ZZ>=cuz 10226   RR+crp 10350   abscabs 11714    ~~> cli 11953
This theorem is referenced by:  clim  11963  climcl  11968  climi  11979  climrlim2  12016  fclim  12022  climrecl  12052  climge0  12053  iserex  12125  caurcvg2  12145  caucvg  12146  iseralt  12152  fsumcvg3  12197  cvgcmpce  12271  climfsum  12273  climcnds  12305  trirecip  12316  ovoliunlem1  18856  mbflimlem  19017  abelthlem5  19806  emcllem6  20289  stirlinglem12  27234
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-opab 4080  df-xp 4695  df-rel 4696  df-clim 11957
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