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Theorem limcfval 21306
Description: Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
limcval.j  |-  J  =  ( Kt  ( A  u.  { B } ) )
limcval.k  |-  K  =  ( TopOpen ` fld )
Assertion
Ref Expression
limcfval  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  (
( F lim CC  B
)  =  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) }  /\  ( F lim CC  B )  C_  CC ) )
Distinct variable groups:    y, z, A    y, B, z    y, F, z    y, K, z   
y, J
Allowed substitution hint:    J( z)

Proof of Theorem limcfval
Dummy variables  f 
j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-limc 21300 . . . 4  |- 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
) } )
21a1i 11 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  -> 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
) } ) )
3 fvex 5698 . . . . . 6  |-  ( TopOpen ` fld )  e.  _V
43a1i 11 . . . . 5  |-  ( ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  -> 
( TopOpen ` fld )  e.  _V )
5 simplrl 754 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  f  =  F )
65dmeqd 5038 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  f  =  dom  F )
7 simpll1 1022 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  F : A
--> CC )
8 fdm 5560 . . . . . . . . . 10  |-  ( F : A --> CC  ->  dom 
F  =  A )
97, 8syl 16 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  F  =  A )
106, 9eqtrd 2473 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  f  =  A )
11 simplrr 755 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  x  =  B )
1211sneqd 3886 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  { x }  =  { B } )
1310, 12uneq12d 3508 . . . . . . 7  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( dom  f  u.  { x } )  =  ( A  u.  { B } ) )
1411eqeq2d 2452 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( z  =  x  <->  z  =  B ) )
155fveq1d 5690 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( f `  z )  =  ( F `  z ) )
1614, 15ifbieq2d 3811 . . . . . . 7  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  if (
z  =  x ,  y ,  ( f `
 z ) )  =  if ( z  =  B ,  y ,  ( F `  z ) ) )
1713, 16mpteq12dv 4367 . . . . . 6  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( z  e.  ( dom  f  u. 
{ x } ) 
|->  if ( z  =  x ,  y ,  ( f `  z
) ) )  =  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) ) )
18 simpr 458 . . . . . . . . . . 11  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  j  =  ( TopOpen ` fld ) )
19 limcval.k . . . . . . . . . . 11  |-  K  =  ( TopOpen ` fld )
2018, 19syl6eqr 2491 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  j  =  K )
2120, 13oveq12d 6108 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( jt  ( dom  f  u.  { x } ) )  =  ( Kt  ( A  u.  { B } ) ) )
22 limcval.j . . . . . . . . 9  |-  J  =  ( Kt  ( A  u.  { B } ) )
2321, 22syl6eqr 2491 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( jt  ( dom  f  u.  { x } ) )  =  J )
2423, 20oveq12d 6108 . . . . . . 7  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( (
jt  ( dom  f  u. 
{ x } ) )  CnP  j )  =  ( J  CnP  K ) )
2524, 11fveq12d 5694 . . . . . 6  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( (
( jt  ( dom  f  u.  { x } ) )  CnP  j ) `
 x )  =  ( ( J  CnP  K ) `  B ) )
2617, 25eleq12d 2509 . . . . 5  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( (
z  e.  ( dom  f  u.  { x } )  |->  if ( z  =  x ,  y ,  ( f `
 z ) ) )  e.  ( ( ( jt  ( dom  f  u.  { x } ) )  CnP  j ) `
 x )  <->  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) ) )
274, 26sbcied 3220 . . . 4  |-  ( ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  -> 
( [. ( TopOpen ` fld )  /  j ]. ( z  e.  ( dom  f  u.  {
x } )  |->  if ( z  =  x ,  y ,  ( f `  z ) ) )  e.  ( ( ( jt  ( dom  f  u.  { x } ) )  CnP  j ) `  x
)  <->  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) ) )
2827abbidv 2555 . . 3  |-  ( ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  ->  { 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
) }  =  {
y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )  e.  ( ( J  CnP  K
) `  B ) } )
29 cnex 9359 . . . . 5  |-  CC  e.  _V
30 elpm2r 7226 . . . . 5  |-  ( ( ( CC  e.  _V  /\  CC  e.  _V )  /\  ( F : A --> CC  /\  A  C_  CC ) )  ->  F  e.  ( CC  ^pm  CC ) )
3129, 29, 30mpanl12 677 . . . 4  |-  ( ( F : A --> CC  /\  A  C_  CC )  ->  F  e.  ( CC  ^pm 
CC ) )
32313adant3 1003 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  F  e.  ( CC  ^pm  CC ) )
33 simp3 985 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  B  e.  CC )
34 eqid 2441 . . . . . 6  |-  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )  =  ( z  e.  ( A  u.  { B }
)  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )
3522, 19, 34limcvallem 21305 . . . . 5  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  (
( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B )  -> 
y  e.  CC ) )
3635abssdv 3423 . . . 4  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) } 
C_  CC )
3729ssex 4433 . . . 4  |-  ( { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )  e.  ( ( J  CnP  K
) `  B ) }  C_  CC  ->  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) }  e.  _V )
3836, 37syl 16 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) }  e.  _V )
392, 28, 32, 33, 38ovmpt2d 6217 . 2  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  ( F lim CC  B )  =  { y  |  ( z  e.  ( A  u.  { B }
)  |->  if ( z  =  B ,  y ,  ( F `  z ) ) )  e.  ( ( J  CnP  K ) `  B ) } )
4039, 36eqsstrd 3387 . 2  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  ( F lim CC  B )  C_  CC )
4139, 40jca 529 1  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  (
( F lim CC  B
)  =  { y  |  ( z  e.  ( A  u.  { B } )  |->  if ( z  =  B , 
y ,  ( F `
 z ) ) )  e.  ( ( J  CnP  K ) `
 B ) }  /\  ( F lim CC  B )  C_  CC ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   {cab 2427   _Vcvv 2970   [.wsbc 3183    u. cun 3323    C_ wss 3325   ifcif 3788   {csn 3874    e. cmpt 4347   dom cdm 4836   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092    ^pm cpm 7211   CCcc 9276   ↾t crest 14355   TopOpenctopn 14356  ℂfldccnfld 17777    CnP ccnp 18788   lim CC climc 21296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-rest 14357  df-topn 14358  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cnp 18791  df-xms 19854  df-ms 19855  df-limc 21300
This theorem is referenced by:  ellimc  21307  limccl  21309
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