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Theorem limcrcl 14212
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 14210 . . 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 6071 . 2  |-  ( C  e.  ( F lim CC  B )  ->  ( F  e.  ( CC  ^pm 
CC )  /\  B  e.  CC ) )
3 cnex 7937 . . . . 5  |-  CC  e.  _V
43, 3elpm2 6682 . . . 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 3131   class class class wbr 4005   dom cdm 4628   -->wf 5214   ` cfv 5218  (class class class)co 5877    ^pm cpm 6651   CCcc 7811    < clt 7994    - cmin 8130   # cap 8540   RR+crp 9655   abscabs 11008   lim CC climc 14208
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904
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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pm 6653  df-limced 14210
This theorem is referenced by:  limccl  14213  limcdifap  14216  limcimolemlt  14218  limcresi  14220  limccnpcntop  14229  limccnp2lem  14230  limccnp2cntop  14231  limccoap  14232
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