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Theorem climrel 11188
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 11187 . 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 4714 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 2128   A.wral 2435   E.wrex 2436   class class class wbr 3967   Rel wrel 4593   ` cfv 5172  (class class class)co 5826   CCcc 7732    < clt 7914    - cmin 8050   ZZcz 9172   ZZ>=cuz 9444   RR+crp 9566   abscabs 10908    ~~> cli 11186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-opab 4028  df-xp 4594  df-rel 4595  df-clim 11187
This theorem is referenced by:  clim  11189  climcl  11190  climi  11195  fclim  11202  climrecl  11232  iserex  11247  climrecvg1n  11256  climcvg1nlem  11257  fsum3cvg3  11304  trirecip  11409  ntrivcvgap0  11457
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