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Theorem rlimpm 12977
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 12966 . . . . 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 4910 . . . . 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 3385 . . . 4  |-  ~~> r  C_  ( ( CC  ^pm  RR )  X.  CC )
4 dmss 5038 . . . 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 5268 . . 3  |-  dom  (
( CC  ^pm  RR )  X.  CC )  C_  ( CC  ^pm  RR )
75, 6sstri 3364 . 2  |-  dom  ~~> r  C_  ( CC  ^pm  RR )
8 rlimrel 12970 . . 3  |-  Rel  ~~> r
98releldmi 5075 . 2  |-  ( F  ~~> r  A  ->  F  e.  dom  ~~> r  )
107, 9sseldi 3353 1  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2714   E.wrex 2715    C_ wss 3327   class class class wbr 4291   {copab 4348    X. cxp 4837   dom cdm 4839   ` cfv 5417  (class class class)co 6090    ^pm cpm 7214   CCcc 9279   RRcr 9280    < clt 9417    <_ cle 9418    - cmin 9594   RR+crp 10990   abscabs 12722    ~~> r crli 12962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rlim 12966
This theorem is referenced by:  rlimf  12978  rlimss  12979  rlimclim1  13022
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