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Theorem limcfval 19222
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 19216 . . . 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 10 . . 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 5539 . . . . . 6  |-  ( TopOpen ` fld )  e.  _V
43a1i 10 . . . . 5  |-  ( ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  -> 
( TopOpen ` fld )  e.  _V )
5 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  f  =  F )
65dmeqd 4881 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  f  =  dom  F )
7 simpll1 994 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  F : A
--> CC )
8 fdm 5393 . . . . . . . . . 10  |-  ( F : A --> CC  ->  dom 
F  =  A )
97, 8syl 15 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  F  =  A )
106, 9eqtrd 2315 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  dom  f  =  A )
11 simplrr 737 . . . . . . . . 9  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  x  =  B )
1211sneqd 3653 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  { x }  =  { B } )
1310, 12uneq12d 3330 . . . . . . 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 2294 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( z  =  x  <->  z  =  B ) )
155fveq1d 5527 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( f `  z )  =  ( F `  z ) )
1614, 15ifbieq2d 3585 . . . . . . 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 4098 . . . . . 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 447 . . . . . . . . . . 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 2333 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  j  =  K )
2120, 13oveq12d 5876 . . . . . . . . 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 2333 . . . . . . . 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 5876 . . . . . . 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 5531 . . . . . 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 2351 . . . . 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 3027 . . . 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 2397 . . 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 8818 . . . . 5  |-  CC  e.  _V
30 elpm2r 6788 . . . . 5  |-  ( ( ( CC  e.  _V  /\  CC  e.  _V )  /\  ( F : A --> CC  /\  A  C_  CC ) )  ->  F  e.  ( CC  ^pm  CC ) )
3129, 29, 30mpanl12 663 . . . 4  |-  ( ( F : A --> CC  /\  A  C_  CC )  ->  F  e.  ( CC  ^pm 
CC ) )
32313adant3 975 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  F  e.  ( CC  ^pm  CC ) )
33 simp3 957 . . 3  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  B  e.  CC )
34 eqid 2283 . . . . . 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 19221 . . . . 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 3247 . . . 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 4158 . . . 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 15 . . 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 5975 . 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 3212 . 2  |-  ( ( F : A --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  ( F lim CC  B )  C_  CC )
4139, 40jca 518 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   [.wsbc 2991    u. cun 3150    C_ wss 3152   ifcif 3565   {csn 3640    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ^pm cpm 6773   CCcc 8735   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377    CnP ccnp 16955   lim CC climc 19212
This theorem is referenced by:  ellimc  19223  limccl  19225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cnp 16958  df-xms 17885  df-ms 17886  df-limc 19216
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