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Theorem limcdifap 12512
Description: It suffices to consider functions which are not defined at 
B to define the limit of a function. In particular, the value of the original function  F at  B does not affect the limit of  F. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
Hypotheses
Ref Expression
limccl.f  |-  ( ph  ->  F : A --> CC )
limcdifap.a  |-  ( ph  ->  A  C_  CC )
Assertion
Ref Expression
limcdifap  |-  ( ph  ->  ( F lim CC  B
)  =  ( ( F  |`  { x  e.  A  |  x #  B } ) lim CC  B
) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    F( x)

Proof of Theorem limcdifap
Dummy variables  d  e  u  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limcrcl 12509 . . . . 5  |-  ( u  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC ) )
21simp3d 963 . . . 4  |-  ( u  e.  ( F lim CC  B )  ->  B  e.  CC )
32a1i 9 . . 3  |-  ( ph  ->  ( u  e.  ( F lim CC  B )  ->  B  e.  CC ) )
4 limcrcl 12509 . . . . 5  |-  ( u  e.  ( ( F  |`  { x  e.  A  |  x #  B }
) lim CC  B )  ->  ( ( F  |`  { x  e.  A  |  x #  B }
) : dom  ( F  |`  { x  e.  A  |  x #  B } ) --> CC  /\  dom  ( F  |`  { x  e.  A  |  x #  B } )  C_  CC  /\  B  e.  CC ) )
54simp3d 963 . . . 4  |-  ( u  e.  ( ( F  |`  { x  e.  A  |  x #  B }
) lim CC  B )  ->  B  e.  CC )
65a1i 9 . . 3  |-  ( ph  ->  ( u  e.  ( ( F  |`  { x  e.  A  |  x #  B } ) lim CC  B
)  ->  B  e.  CC ) )
7 breq1 3878 . . . . . . . . . . . . . . . . 17  |-  ( x  =  z  ->  (
x #  B  <->  z #  B
) )
8 simplr 500 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A
)  /\  z #  B
)  ->  z  e.  A )
9 simpr 109 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A
)  /\  z #  B
)  ->  z #  B
)
107, 8, 9elrabd 2795 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A
)  /\  z #  B
)  ->  z  e.  { x  e.  A  |  x #  B } )
11 fvres 5377 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  { x  e.  A  |  x #  B }  ->  ( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  =  ( F `  z ) )
1211eqcomd 2105 . . . . . . . . . . . . . . . 16  |-  ( z  e.  { x  e.  A  |  x #  B }  ->  ( F `  z )  =  ( ( F  |`  { x  e.  A  |  x #  B } ) `  z
) )
1310, 12syl 14 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A
)  /\  z #  B
)  ->  ( F `  z )  =  ( ( F  |`  { x  e.  A  |  x #  B } ) `  z
) )
1413fvoveq1d 5728 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A
)  /\  z #  B
)  ->  ( abs `  ( ( F `  z )  -  u
) )  =  ( abs `  ( ( ( F  |`  { x  e.  A  |  x #  B } ) `  z
)  -  u ) ) )
1514breq1d 3885 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A
)  /\  z #  B
)  ->  ( ( abs `  ( ( F `
 z )  -  u ) )  < 
e  <->  ( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) )
1615imbi2d 229 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A
)  /\  z #  B
)  ->  ( (
( abs `  (
z  -  B ) )  <  d  -> 
( abs `  (
( F `  z
)  -  u ) )  <  e )  <-> 
( ( abs `  (
z  -  B ) )  <  d  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) )
1716pm5.74da 435 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A )  ->  (
( z #  B  -> 
( ( abs `  (
z  -  B ) )  <  d  -> 
( abs `  (
( F `  z
)  -  u ) )  <  e ) )  <->  ( z #  B  ->  ( ( abs `  (
z  -  B ) )  <  d  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) ) )
18 impexp 261 . . . . . . . . . . 11  |-  ( ( ( z #  B  /\  ( abs `  ( z  -  B ) )  <  d )  -> 
( abs `  (
( F `  z
)  -  u ) )  <  e )  <-> 
( z #  B  -> 
( ( abs `  (
z  -  B ) )  <  d  -> 
( abs `  (
( F `  z
)  -  u ) )  <  e ) ) )
19 impexp 261 . . . . . . . . . . . . 13  |-  ( ( ( z #  B  /\  ( abs `  ( z  -  B ) )  <  d )  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e )  <->  ( z #  B  ->  ( ( abs `  ( z  -  B
) )  <  d  ->  ( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) )
2019imbi2i 225 . . . . . . . . . . . 12  |-  ( ( z #  B  ->  (
( z #  B  /\  ( abs `  ( z  -  B ) )  <  d )  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) )  <->  ( z #  B  ->  ( z #  B  ->  ( ( abs `  (
z  -  B ) )  <  d  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) ) )
21 pm5.4 248 . . . . . . . . . . . 12  |-  ( ( z #  B  ->  (
z #  B  ->  (
( abs `  (
z  -  B ) )  <  d  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) )  <-> 
( z #  B  -> 
( ( abs `  (
z  -  B ) )  <  d  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) )
2220, 21bitri 183 . . . . . . . . . . 11  |-  ( ( z #  B  ->  (
( z #  B  /\  ( abs `  ( z  -  B ) )  <  d )  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) )  <->  ( z #  B  ->  ( ( abs `  ( z  -  B
) )  <  d  ->  ( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) )
2317, 18, 223bitr4g 222 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  e.  CC )  /\  z  e.  A )  ->  (
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( F `  z
)  -  u ) )  <  e )  <-> 
( z #  B  -> 
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) ) )
2423ralbidva 2392 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  CC )  ->  ( A. z  e.  A  (
( z #  B  /\  ( abs `  ( z  -  B ) )  <  d )  -> 
( abs `  (
( F `  z
)  -  u ) )  <  e )  <->  A. z  e.  A  ( z #  B  ->  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  <  d )  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) ) )
257ralrab 2798 . . . . . . . . 9  |-  ( A. z  e.  { x  e.  A  |  x #  B }  ( (
z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( ( F  |`  { x  e.  A  |  x #  B } ) `  z
)  -  u ) )  <  e )  <->  A. z  e.  A  ( z #  B  ->  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  <  d )  -> 
( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) )
2624, 25syl6bbr 197 . . . . . . . 8  |-  ( (
ph  /\  B  e.  CC )  ->  ( A. z  e.  A  (
( z #  B  /\  ( abs `  ( z  -  B ) )  <  d )  -> 
( abs `  (
( F `  z
)  -  u ) )  <  e )  <->  A. z  e.  { x  e.  A  |  x #  B }  ( (
z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( ( F  |`  { x  e.  A  |  x #  B } ) `  z
)  -  u ) )  <  e ) ) )
2726rexbidv 2397 . . . . . . 7  |-  ( (
ph  /\  B  e.  CC )  ->  ( E. d  e.  RR+  A. z  e.  A  ( (
z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( F `
 z )  -  u ) )  < 
e )  <->  E. d  e.  RR+  A. z  e. 
{ x  e.  A  |  x #  B } 
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) )
2827ralbidv 2396 . . . . . 6  |-  ( (
ph  /\  B  e.  CC )  ->  ( A. e  e.  RR+  E. d  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B
) )  <  d
)  ->  ( abs `  ( ( F `  z )  -  u
) )  <  e
)  <->  A. e  e.  RR+  E. d  e.  RR+  A. z  e.  { x  e.  A  |  x #  B } 
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) )
2928anbi2d 455 . . . . 5  |-  ( (
ph  /\  B  e.  CC )  ->  ( ( u  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B
) )  <  d
)  ->  ( abs `  ( ( F `  z )  -  u
) )  <  e
) )  <->  ( u  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e.  {
x  e.  A  |  x #  B }  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
d )  ->  ( abs `  ( ( ( F  |`  { x  e.  A  |  x #  B } ) `  z
)  -  u ) )  <  e ) ) ) )
30 limccl.f . . . . . . 7  |-  ( ph  ->  F : A --> CC )
3130adantr 272 . . . . . 6  |-  ( (
ph  /\  B  e.  CC )  ->  F : A
--> CC )
32 limcdifap.a . . . . . . 7  |-  ( ph  ->  A  C_  CC )
3332adantr 272 . . . . . 6  |-  ( (
ph  /\  B  e.  CC )  ->  A  C_  CC )
34 simpr 109 . . . . . 6  |-  ( (
ph  /\  B  e.  CC )  ->  B  e.  CC )
3531, 33, 34ellimc3ap 12511 . . . . 5  |-  ( (
ph  /\  B  e.  CC )  ->  ( u  e.  ( F lim CC  B )  <->  ( u  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( F `  z
)  -  u ) )  <  e ) ) ) )
36 ssrab2 3129 . . . . . . 7  |-  { x  e.  A  |  x #  B }  C_  A
37 fssres 5234 . . . . . . 7  |-  ( ( F : A --> CC  /\  { x  e.  A  |  x #  B }  C_  A
)  ->  ( F  |` 
{ x  e.  A  |  x #  B }
) : { x  e.  A  |  x #  B } --> CC )
3831, 36, 37sylancl 407 . . . . . 6  |-  ( (
ph  /\  B  e.  CC )  ->  ( F  |`  { x  e.  A  |  x #  B }
) : { x  e.  A  |  x #  B } --> CC )
3936, 33syl5ss 3058 . . . . . 6  |-  ( (
ph  /\  B  e.  CC )  ->  { x  e.  A  |  x #  B }  C_  CC )
4038, 39, 34ellimc3ap 12511 . . . . 5  |-  ( (
ph  /\  B  e.  CC )  ->  ( u  e.  ( ( F  |`  { x  e.  A  |  x #  B }
) lim CC  B )  <->  ( u  e.  CC  /\  A. e  e.  RR+  E. d  e.  RR+  A. z  e. 
{ x  e.  A  |  x #  B } 
( ( z #  B  /\  ( abs `  (
z  -  B ) )  <  d )  ->  ( abs `  (
( ( F  |`  { x  e.  A  |  x #  B }
) `  z )  -  u ) )  < 
e ) ) ) )
4129, 35, 403bitr4d 219 . . . 4  |-  ( (
ph  /\  B  e.  CC )  ->  ( u  e.  ( F lim CC  B )  <->  u  e.  ( ( F  |`  { x  e.  A  |  x #  B }
) lim CC  B )
) )
4241ex 114 . . 3  |-  ( ph  ->  ( B  e.  CC  ->  ( u  e.  ( F lim CC  B )  <-> 
u  e.  ( ( F  |`  { x  e.  A  |  x #  B } ) lim CC  B
) ) ) )
433, 6, 42pm5.21ndd 662 . 2  |-  ( ph  ->  ( u  e.  ( F lim CC  B )  <-> 
u  e.  ( ( F  |`  { x  e.  A  |  x #  B } ) lim CC  B
) ) )
4443eqrdv 2098 1  |-  ( ph  ->  ( F lim CC  B
)  =  ( ( F  |`  { x  e.  A  |  x #  B } ) lim CC  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   {crab 2379    C_ wss 3021   class class class wbr 3875   dom cdm 4477    |` cres 4479   -->wf 5055   ` cfv 5059  (class class class)co 5706   CCcc 7498    < clt 7672    - cmin 7804   # cap 8209   RR+crp 9291   abscabs 10609   lim CC climc 12505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pm 6475  df-limced 12507
This theorem is referenced by: (None)
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