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Theorem rlimpm 12070
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 12059 . . . . 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 4844 . . . . 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 3284 . . . 4  |-  ~~> r  C_  ( ( CC  ^pm  RR )  X.  CC )
4 dmss 4960 . . . 4  |-  (  ~~> r  C_  ( ( CC  ^pm  RR )  X.  CC )  ->  dom  ~~> r  C_  dom  ( ( CC  ^pm  RR )  X.  CC ) )
53, 4ax-mp 8 . . 3  |-  dom  ~~> r  C_  dom  ( ( CC  ^pm  RR )  X.  CC )
6 dmxpss 5189 . . 3  |-  dom  (
( CC  ^pm  RR )  X.  CC )  C_  ( CC  ^pm  RR )
75, 6sstri 3264 . 2  |-  dom  ~~> r  C_  ( CC  ^pm  RR )
8 rlimrel 12063 . . 3  |-  Rel  ~~> r
98releldmi 4997 . 2  |-  ( F  ~~> r  A  ->  F  e.  dom  ~~> r  )
107, 9sseldi 3254 1  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   A.wral 2619   E.wrex 2620    C_ wss 3228   class class class wbr 4104   {copab 4157    X. cxp 4769   dom cdm 4771   ` 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:  rlimf  12071  rlimss  12072  rlimclim1  12115
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-eu 2213  df-mo 2214  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-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-dm 4781  df-rlim 12059
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