MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimrel Unicode version

Theorem rlimrel 11961
Description: The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
Assertion
Ref Expression
rlimrel  |-  Rel  ~~> r

Proof of Theorem rlimrel
StepHypRef Expression
1 df-rlim 11957 . 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 4810 1  |-  Rel  ~~> r
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1688   A.wral 2544   E.wrex 2545   class class class wbr 4024   dom cdm 4688   Rel wrel 4693   ` cfv 5221  (class class class)co 5819    ^pm cpm 6768   CCcc 8730   RRcr 8731    < clt 8862    <_ cle 8863    - cmin 9032   RR+crp 10349   abscabs 11713    ~~> r crli 11953
This theorem is referenced by:  rlim  11963  rlimpm  11968  rlimdm  12019  caucvgrlem2  12141  caucvgr  12142
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-opab 4079  df-xp 4694  df-rel 4695  df-rlim 11957
  Copyright terms: Public domain W3C validator