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Theorem limcrcl 19240
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 19232 . . 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 6077 . 2  |-  ( C  e.  ( F lim CC  B )  ->  ( F  e.  ( CC  ^pm 
CC )  /\  B  e.  CC ) )
3 cnex 8834 . . . . 5  |-  CC  e.  _V
43, 3elpm2 6815 . . . 4  |-  ( F  e.  ( CC  ^pm  CC )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  CC ) )
54anbi1i 676 . . 3  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC )  <->  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  B  e.  CC ) )
6 df-3an 936 . . 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 243 . 2  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC )  <->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
82, 7sylib 188 1  |-  ( C  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   [.wsbc 3004    u. cun 3163    C_ wss 3165   ifcif 3578   {csn 3653    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393    CnP ccnp 16971   lim CC climc 19228
This theorem is referenced by:  limccl  19241  limcdif  19242  limcresi  19251  limcres  19252  limccnp  19257  limccnp2  19258  limcco  19259  limcun  19261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791  df-limc 19232
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