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Theorem limccl 13697
Description: Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Assertion
Ref Expression
limccl  |-  ( F lim
CC  B )  C_  CC

Proof of Theorem limccl
Dummy variables  d  e  f  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4  |-  ( w  e.  ( F lim CC  B )  ->  w  e.  ( F lim CC  B
) )
2 df-limced 13694 . . . . . 6  |- 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 ) ) ) } )
32elmpocl1 6060 . . . . 5  |-  ( w  e.  ( F lim CC  B )  ->  F  e.  ( CC  ^pm  CC ) )
4 limcrcl 13696 . . . . . 6  |-  ( w  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
54simp3d 1011 . . . . 5  |-  ( w  e.  ( F lim CC  B )  ->  B  e.  CC )
6 cnex 7910 . . . . . . 7  |-  CC  e.  _V
76rabex 4142 . . . . . 6  |-  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) }  e.  _V
87a1i 9 . . . . 5  |-  ( w  e.  ( F lim CC  B )  ->  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) }  e.  _V )
9 simpl 109 . . . . . . . . . 10  |-  ( ( f  =  F  /\  x  =  B )  ->  f  =  F )
109dmeqd 4822 . . . . . . . . . 10  |-  ( ( f  =  F  /\  x  =  B )  ->  dom  f  =  dom  F )
119, 10feq12d 5347 . . . . . . . . 9  |-  ( ( f  =  F  /\  x  =  B )  ->  ( f : dom  f
--> CC  <->  F : dom  F --> CC ) )
1210sseq1d 3182 . . . . . . . . 9  |-  ( ( f  =  F  /\  x  =  B )  ->  ( dom  f  C_  CC 
<->  dom  F  C_  CC ) )
1311, 12anbi12d 473 . . . . . . . 8  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( f : dom  f --> CC  /\  dom  f  C_  CC )  <-> 
( F : dom  F --> CC  /\  dom  F  C_  CC ) ) )
14 simpr 110 . . . . . . . . . 10  |-  ( ( f  =  F  /\  x  =  B )  ->  x  =  B )
1514eleq1d 2244 . . . . . . . . 9  |-  ( ( f  =  F  /\  x  =  B )  ->  ( x  e.  CC  <->  B  e.  CC ) )
1614breq2d 4010 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  x  =  B )  ->  ( z #  x  <->  z #  B
) )
1714oveq2d 5881 . . . . . . . . . . . . . . . 16  |-  ( ( f  =  F  /\  x  =  B )  ->  ( z  -  x
)  =  ( z  -  B ) )
1817fveq2d 5511 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  x  =  B )  ->  ( abs `  (
z  -  x ) )  =  ( abs `  ( z  -  B
) ) )
1918breq1d 4008 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( abs `  (
z  -  x ) )  <  d  <->  ( abs `  ( z  -  B
) )  <  d
) )
2016, 19anbi12d 473 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( z #  x  /\  ( abs `  (
z  -  x ) )  <  d )  <-> 
( z #  B  /\  ( abs `  ( z  -  B ) )  <  d ) ) )
219fveq1d 5509 . . . . . . . . . . . . . . 15  |-  ( ( f  =  F  /\  x  =  B )  ->  ( f `  z
)  =  ( F `
 z ) )
2221fvoveq1d 5887 . . . . . . . . . . . . . 14  |-  ( ( f  =  F  /\  x  =  B )  ->  ( abs `  (
( f `  z
)  -  y ) )  =  ( abs `  ( ( F `  z )  -  y
) ) )
2322breq1d 4008 . . . . . . . . . . . . 13  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( abs `  (
( f `  z
)  -  y ) )  <  e  <->  ( abs `  ( ( F `  z )  -  y
) )  <  e
) )
2420, 23imbi12d 234 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( ( z #  x  /\  ( abs `  ( z  -  x
) )  <  d
)  ->  ( abs `  ( ( f `  z )  -  y
) )  <  e
)  <->  ( ( z #  B  /\  ( abs `  ( z  -  B
) )  <  d
)  ->  ( abs `  ( ( F `  z )  -  y
) )  <  e
) ) )
2510, 24raleqbidv 2682 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  x  =  B )  ->  ( A. z  e. 
dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
d )  ->  ( abs `  ( ( f `
 z )  -  y ) )  < 
e )  <->  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) )
2625rexbidv 2476 . . . . . . . . . 10  |-  ( ( f  =  F  /\  x  =  B )  ->  ( E. d  e.  RR+  A. z  e.  dom  f ( ( z #  x  /\  ( abs `  ( z  -  x
) )  <  d
)  ->  ( abs `  ( ( f `  z )  -  y
) )  <  e
)  <->  E. d  e.  RR+  A. z  e.  dom  F
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( F `  z
)  -  y ) )  <  e ) ) )
2726ralbidv 2475 . . . . . . . . 9  |-  ( ( f  =  F  /\  x  =  B )  ->  ( A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  f
( ( z #  x  /\  ( abs `  (
z  -  x ) )  <  d )  ->  ( abs `  (
( f `  z
)  -  y ) )  <  e )  <->  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) )
2815, 27anbi12d 473 . . . . . . . 8  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( 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
) )  <->  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( F `  z
)  -  y ) )  <  e ) ) ) )
2913, 28anbi12d 473 . . . . . . 7  |-  ( ( f  =  F  /\  x  =  B )  ->  ( ( ( 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 ) ) )  <-> 
( ( F : dom  F --> CC  /\  dom  F 
C_  CC )  /\  ( B  e.  CC  /\ 
A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) ) )
3029rabbidv 2724 . . . . . 6  |-  ( ( f  =  F  /\  x  =  B )  ->  { 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 ) ) ) }  =  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) } )
3130, 2ovmpoga 5994 . . . . 5  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  B  e.  CC  /\  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) }  e.  _V )  ->  ( F lim CC  B
)  =  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) } )
323, 5, 8, 31syl3anc 1238 . . . 4  |-  ( w  e.  ( F lim CC  B )  ->  ( F lim CC  B )  =  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\ 
A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) } )
331, 32eleqtrd 2254 . . 3  |-  ( w  e.  ( F lim CC  B )  ->  w  e.  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\ 
A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) } )
34 elrabi 2888 . . 3  |-  ( w  e.  { y  e.  CC  |  ( ( F : dom  F --> CC  /\  dom  F  C_  CC )  /\  ( B  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
dom  F ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  y ) )  < 
e ) ) ) }  ->  w  e.  CC )
3533, 34syl 14 . 2  |-  ( w  e.  ( F lim CC  B )  ->  w  e.  CC )
3635ssriv 3157 1  |-  ( F lim
CC  B )  C_  CC
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2146   A.wral 2453   E.wrex 2454   {crab 2457   _Vcvv 2735    C_ wss 3127   class class class wbr 3998   dom cdm 4620   -->wf 5204   ` cfv 5208  (class class class)co 5865    ^pm cpm 6639   CCcc 7784    < clt 7966    - cmin 8102   # cap 8512   RR+crp 9622   abscabs 10972   lim CC climc 13692
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pm 6641  df-limced 13694
This theorem is referenced by:  reldvg  13717  dvfvalap  13719  dvcl  13721
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