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Theorem climrel 10257
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 10256 . 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 4491 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1434   A.wral 2349   E.wrex 2350   class class class wbr 3793   Rel wrel 4376   ` cfv 4932  (class class class)co 5543   CCcc 7041    < clt 7215    - cmin 7346   ZZcz 8432   ZZ>=cuz 8700   RR+crp 8815   abscabs 10021    ~~> cli 10255
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848  df-xp 4377  df-rel 4378  df-clim 10256
This theorem is referenced by:  clim  10258  climcl  10259  climi  10264  fclim  10271  climrecl  10300  iiserex  10315  climrecvg1n  10323  climcvg1nlem  10324
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