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Theorem limcfval 19238
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 19232 . . . 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 5555 . . . . . 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 4897 . . . . . . . . 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 5409 . . . . . . . . . 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 2328 . . . . . . . 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 3666 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  { x }  =  { B } )
1310, 12uneq12d 3343 . . . . . . 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 2307 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( z  =  x  <->  z  =  B ) )
155fveq1d 5543 . . . . . . . 8  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  ( f `  z )  =  ( F `  z ) )
1614, 15ifbieq2d 3598 . . . . . . 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 4114 . . . . . 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 2346 . . . . . . . . . 10  |-  ( ( ( ( F : A
--> CC  /\  A  C_  CC  /\  B  e.  CC )  /\  ( f  =  F  /\  x  =  B ) )  /\  j  =  ( TopOpen ` fld )
)  ->  j  =  K )
2120, 13oveq12d 5892 . . . . . . . . 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 2346 . . . . . . . 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 5892 . . . . . . 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 5547 . . . . . 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 2364 . . . . 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 3040 . . . 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 2410 . . 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 8834 . . . . 5  |-  CC  e.  _V
30 elpm2r 6804 . . . . 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 2296 . . . . . 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 19237 . . . . 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 3260 . . . 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 4174 . . . 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 5991 . 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 3225 . 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 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   [.wsbc 3004    u. cun 3163    C_ wss 3165   ifcif 3578   {csn 3653    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^pm cpm 6789   CCcc 8751   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393    CnP ccnp 16971   lim CC climc 19228
This theorem is referenced by:  ellimc  19239  limccl  19241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cnp 16974  df-xms 17901  df-ms 17902  df-limc 19232
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