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Theorem rlimrel 12275
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 12271 . 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 4991 1  |-  Rel  ~~> r
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   dom cdm 4869   Rel wrel 4874   ` cfv 5445  (class class class)co 6072    ^pm cpm 7010   CCcc 8977   RRcr 8978    < clt 9109    <_ cle 9110    - cmin 9280   RR+crp 10601   abscabs 12027    ~~> r crli 12267
This theorem is referenced by:  rlim  12277  rlimpm  12282  rlimdm  12333  caucvgrlem2  12456  caucvgr  12457  rlimdmafv  27955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4875  df-rel 4876  df-rlim 12271
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