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Theorem limcrcl 22363
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 22355 . . 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 6416 . 2  |-  ( C  e.  ( F lim CC  B )  ->  ( F  e.  ( CC  ^pm 
CC )  /\  B  e.  CC ) )
3 cnex 9484 . . . . 5  |-  CC  e.  _V
43, 3elpm2 7369 . . . 4  |-  ( F  e.  ( CC  ^pm  CC )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  CC ) )
54anbi1i 693 . . 3  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC )  <->  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  B  e.  CC ) )
6 df-3an 973 . . 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 367    /\ w3a 971    e. wcel 1826   {cab 2367   [.wsbc 3252    u. cun 3387    C_ wss 3389   ifcif 3857   {csn 3944    |-> cmpt 4425   dom cdm 4913   -->wf 5492   ` cfv 5496  (class class class)co 6196    ^pm cpm 7339   CCcc 9401   ↾t crest 14828   TopOpenctopn 14829  ℂfldccnfld 18533    CnP ccnp 19812   lim CC climc 22351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-pm 7341  df-limc 22355
This theorem is referenced by:  limccl  22364  limcdif  22365  limcresi  22374  limcres  22375  limccnp  22380  limccnp2  22381  limcco  22382  limcun  22384  mullimc  31788  limccog  31792  mullimcf  31795  limcperiod  31800  limcmptdm  31807  neglimc  31819  addlimc  31820  0ellimcdiv  31821  reclimc  31825
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