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Theorem rlimpm 11925
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
StepHypRef Expression
1 df-rlim 11914 . . . . 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 4736 . . . . 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 3169 . . . 4  |-  ~~> r  C_  ( ( CC  ^pm  RR )  X.  CC )
4 dmss 4852 . . . 4  |-  (  ~~> r  C_  ( ( CC  ^pm  RR )  X.  CC )  ->  dom  ~~> r  C_  dom  ( ( CC  ^pm  RR )  X.  CC ) )
53, 4ax-mp 10 . . 3  |-  dom  ~~> r  C_  dom  ( ( CC  ^pm  RR )  X.  CC )
6 dmxpss 5081 . . 3  |-  dom  (
( CC  ^pm  RR )  X.  CC )  C_  ( CC  ^pm  RR )
75, 6sstri 3149 . 2  |-  dom  ~~> r  C_  ( CC  ^pm  RR )
8 rlimrel 11918 . . 3  |-  Rel  ~~> r
98releldmi 4889 . 2  |-  ( F  ~~> r  A  ->  F  e.  dom  ~~> r  )
107, 9sseldi 3139 1  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   A.wral 2516   E.wrex 2517    C_ wss 3113   class class class wbr 3983   {copab 4036    X. cxp 4645   dom cdm 4647   ` cfv 4659  (class class class)co 5778    ^pm cpm 6727   CCcc 8689   RRcr 8690    < clt 8821    <_ cle 8822    - cmin 8991   RR+crp 10307   abscabs 11670    ~~> r crli 11910
This theorem is referenced by:  rlimf  11926  rlimss  11927  rlimclim1  11970
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038  df-xp 4661  df-rel 4662  df-cnv 4663  df-dm 4665  df-rlim 11914
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