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Theorem climrel 11243
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 11242 . 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 4737 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 2141   A.wral 2448   E.wrex 2449   class class class wbr 3989   Rel wrel 4616   ` cfv 5198  (class class class)co 5853   CCcc 7772    < clt 7954    - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   RR+crp 9610   abscabs 10961    ~~> cli 11241
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618  df-clim 11242
This theorem is referenced by:  clim  11244  climcl  11245  climi  11250  fclim  11257  climrecl  11287  iserex  11302  climrecvg1n  11311  climcvg1nlem  11312  fsum3cvg3  11359  trirecip  11464  ntrivcvgap0  11512
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