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Theorem rlimpm 13302
Description: Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimpm  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )

Proof of Theorem rlimpm
Dummy variables  w  f  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rlim 13291 . . . . 5  |-  ~~> 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 ) ) }
2 opabssxp 5064 . . . . 5  |-  { <. 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 ) ) }  C_  (
( CC  ^pm  RR )  X.  CC )
31, 2eqsstri 3519 . . . 4  |-  ~~> r  C_  ( ( CC  ^pm  RR )  X.  CC )
4 dmss 5192 . . . 4  |-  (  ~~> r  C_  ( ( CC  ^pm  RR )  X.  CC )  ->  dom  ~~> r  C_  dom  ( ( CC  ^pm  RR )  X.  CC ) )
53, 4ax-mp 5 . . 3  |-  dom  ~~> r  C_  dom  ( ( CC  ^pm  RR )  X.  CC )
6 dmxpss 5428 . . 3  |-  dom  (
( CC  ^pm  RR )  X.  CC )  C_  ( CC  ^pm  RR )
75, 6sstri 3498 . 2  |-  dom  ~~> r  C_  ( CC  ^pm  RR )
8 rlimrel 13295 . . 3  |-  Rel  ~~> r
98releldmi 5229 . 2  |-  ( F  ~~> r  A  ->  F  e.  dom  ~~> r  )
107, 9sseldi 3487 1  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   class class class wbr 4437   {copab 4494    X. cxp 4987   dom cdm 4989   ` cfv 5578  (class class class)co 6281    ^pm cpm 7423   CCcc 9493   RRcr 9494    < clt 9631    <_ cle 9632    - cmin 9810   RR+crp 11229   abscabs 13046    ~~> r crli 13287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rlim 13291
This theorem is referenced by:  rlimf  13303  rlimss  13304  rlimclim1  13347
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