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Theorem climrel 10664
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 10663 . 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 4563 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1438   A.wral 2359   E.wrex 2360   class class class wbr 3845   Rel wrel 4443   ` cfv 5015  (class class class)co 5652   CCcc 7346    < clt 7520    - cmin 7651   ZZcz 8748   ZZ>=cuz 9017   RR+crp 9132   abscabs 10426    ~~> cli 10662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-rel 4445  df-clim 10663
This theorem is referenced by:  clim  10665  climcl  10666  climi  10671  fclim  10678  climrecl  10708  iserex  10723  climrecvg1n  10733  climcvg1nlem  10734  fsum3cvg3  10785  trirecip  10891
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