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Theorem rlimrel 12063
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 12059 . 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 4893 1  |-  Rel  ~~> r
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   A.wral 2619   E.wrex 2620   class class class wbr 4104   dom cdm 4771   Rel wrel 4776   ` cfv 5337  (class class class)co 5945    ^pm cpm 6861   CCcc 8825   RRcr 8826    < clt 8957    <_ cle 8958    - cmin 9127   RR+crp 10446   abscabs 11815    ~~> r crli 12055
This theorem is referenced by:  rlim  12065  rlimpm  12070  rlimdm  12121  caucvgrlem2  12244  caucvgr  12245  rlimdmafv  27365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-opab 4159  df-xp 4777  df-rel 4778  df-rlim 12059
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