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Theorem climrel 9774
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 9773 . 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 4463 1  |-  Rel  ~~>
Colors of variables: wff set class
Syntax hints:    /\ wa 97    e. wcel 1393   A.wral 2306   E.wrex 2307   class class class wbr 3764   Rel wrel 4350   ` cfv 4902  (class class class)co 5512   CCcc 6885    < clt 7058    - cmin 7180   ZZcz 8243   ZZ>=cuz 8471   RR+crp 8581   abscabs 9569    ~~> cli 9772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-clim 9773
This theorem is referenced by:  clim  9775  climcl  9776  climi  9781  fclim  9788  climrecl  9817  iiserex  9832  climrecvg1n  9840  climcvg1nlem  9841
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