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Theorem climrel 10989
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 10988 . 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 4633 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 1463   A.wral 2391   E.wrex 2392   class class class wbr 3897   Rel wrel 4512   ` cfv 5091  (class class class)co 5740   CCcc 7582    < clt 7764    - cmin 7897   ZZcz 9005   ZZ>=cuz 9275   RR+crp 9390   abscabs 10709    ~~> cli 10987
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513  df-rel 4514  df-clim 10988
This theorem is referenced by:  clim  10990  climcl  10991  climi  10996  fclim  11003  climrecl  11033  iserex  11048  climrecvg1n  11057  climcvg1nlem  11058  fsum3cvg3  11105  trirecip  11210
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