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Theorem rlimpm 13274
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 13263 . . . . 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 5067 . . . . 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 3529 . . . 4  |-  ~~> r  C_  ( ( CC  ^pm  RR )  X.  CC )
4 dmss 5195 . . . 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 5431 . . 3  |-  dom  (
( CC  ^pm  RR )  X.  CC )  C_  ( CC  ^pm  RR )
75, 6sstri 3508 . 2  |-  dom  ~~> r  C_  ( CC  ^pm  RR )
8 rlimrel 13267 . . 3  |-  Rel  ~~> r
98releldmi 5232 . 2  |-  ( F  ~~> r  A  ->  F  e.  dom  ~~> r  )
107, 9sseldi 3497 1  |-  ( F  ~~> r  A  ->  F  e.  ( CC  ^pm  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   A.wral 2809   E.wrex 2810    C_ wss 3471   class class class wbr 4442   {copab 4499    X. cxp 4992   dom cdm 4994   ` cfv 5581  (class class class)co 6277    ^pm cpm 7413   CCcc 9481   RRcr 9482    < clt 9619    <_ cle 9620    - cmin 9796   RR+crp 11211   abscabs 13019    ~~> r crli 13259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-dm 5004  df-rlim 13263
This theorem is referenced by:  rlimf  13275  rlimss  13276  rlimclim1  13319
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