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Theorem limcrcl 22146
Description: Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)
Assertion
Ref Expression
limcrcl  |-  ( C  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )

Proof of Theorem limcrcl
Dummy variables  f 
j  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-limc 22138 . . 3  |- lim CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e.  CC  |->  { y  |  [. ( TopOpen ` fld )  /  j ]. ( z  e.  ( dom  f  u.  {
x } )  |->  if ( z  =  x ,  y ,  ( f `  z ) ) )  e.  ( ( ( jt  ( dom  f  u.  { x } ) )  CnP  j ) `  x
) } )
21elmpt2cl 6512 . 2  |-  ( C  e.  ( F lim CC  B )  ->  ( F  e.  ( CC  ^pm 
CC )  /\  B  e.  CC ) )
3 cnex 9585 . . . . 5  |-  CC  e.  _V
43, 3elpm2 7462 . . . 4  |-  ( F  e.  ( CC  ^pm  CC )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  CC ) )
54anbi1i 695 . . 3  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC )  <->  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  B  e.  CC ) )
6 df-3an 975 . . 3  |-  ( ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) 
<->  ( ( F : dom  F --> CC  /\  dom  F 
C_  CC )  /\  B  e.  CC )
)
75, 6bitr4i 252 . 2  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC )  <->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
82, 7sylib 196 1  |-  ( C  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   {cab 2452   [.wsbc 3336    u. cun 3479    C_ wss 3481   ifcif 3945   {csn 4033    |-> cmpt 4511   dom cdm 5005   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^pm cpm 7433   CCcc 9502   ↾t crest 14693   TopOpenctopn 14694  ℂfldccnfld 18290    CnP ccnp 19594   lim CC climc 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-pm 7435  df-limc 22138
This theorem is referenced by:  limccl  22147  limcdif  22148  limcresi  22157  limcres  22158  limccnp  22163  limccnp2  22164  limcco  22165  limcun  22167  mullimc  31481  limccog  31485  mullimcf  31488  limcperiod  31493  limcmptdm  31500  neglimc  31512  addlimc  31513  0ellimcdiv  31514  reclimc  31518
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