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Theorem rlimrel 11967
Description: The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
Assertion
Ref Expression
rlimrel  |-  Rel  ~~> r

Proof of Theorem rlimrel
Dummy variables  w  x  y  z  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rlim 11963 . 2  |-  ~~> r  =  { <. f ,  x >.  |  ( ( f  e.  ( CC  ^pm  RR )  /\  x  e.  CC )  /\  A. y  e.  RR+  E. z  e.  RR  A. w  e. 
dom  f ( z  <_  w  ->  ( abs `  ( ( f `
 w )  -  x ) )  < 
y ) ) }
21relopabi 4811 1  |-  Rel  ~~> r
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   dom cdm 4689   Rel wrel 4694   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   RRcr 8736    < clt 8867    <_ cle 8868    - cmin 9037   RR+crp 10354   abscabs 11719    ~~> r crli 11959
This theorem is referenced by:  rlim  11969  rlimpm  11974  rlimdm  12025  caucvgrlem2  12147  caucvgr  12148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-rlim 11963
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