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Theorem limcrcl 14063
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  d  e  f  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-limced 14061 . . 3  |- lim CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e.  CC  |->  { y  e.  CC  |  ( ( f : dom  f --> CC 
/\  dom  f  C_  CC )  /\  (
x  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
d )  ->  ( abs `  ( ( f `
 z )  -  y ) )  < 
e ) ) ) } )
21elmpocl 6068 . 2  |-  ( C  e.  ( F lim CC  B )  ->  ( F  e.  ( CC  ^pm 
CC )  /\  B  e.  CC ) )
3 cnex 7934 . . . . 5  |-  CC  e.  _V
43, 3elpm2 6679 . . . 4  |-  ( F  e.  ( CC  ^pm  CC )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  CC ) )
54anbi1i 458 . . 3  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC )  <->  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  B  e.  CC ) )
6 df-3an 980 . . 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 187 . 2  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC )  <->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
82, 7sylib 122 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 104    /\ w3a 978    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459    C_ wss 3129   class class class wbr 4003   dom cdm 4626   -->wf 5212   ` cfv 5216  (class class class)co 5874    ^pm cpm 6648   CCcc 7808    < clt 7991    - cmin 8127   # cap 8537   RR+crp 9652   abscabs 11005   lim CC climc 14059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pm 6650  df-limced 14061
This theorem is referenced by:  limccl  14064  limcdifap  14067  limcimolemlt  14069  limcresi  14071  limccnpcntop  14080  limccnp2lem  14081  limccnp2cntop  14082  limccoap  14083
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