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Theorem flfcntr 21136
Description: A continuous function's value is always in the trace of its filter limit. (Contributed by Thierry Arnoux, 30-Aug-2020.)
Hypotheses
Ref Expression
flfcntr.c  |-  C  = 
U. J
flfcntr.b  |-  B  = 
U. K
flfcntr.j  |-  ( ph  ->  J  e.  Top )
flfcntr.a  |-  ( ph  ->  A  C_  C )
flfcntr.1  |-  ( ph  ->  F  e.  ( ( Jt  A )  Cn  K
) )
flfcntr.y  |-  ( ph  ->  X  e.  A )
Assertion
Ref Expression
flfcntr  |-  ( ph  ->  ( F `  X
)  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { X } )t  A ) ) `  F ) )

Proof of Theorem flfcntr
Dummy variables  a  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flfcntr.1 . . . . 5  |-  ( ph  ->  F  e.  ( ( Jt  A )  Cn  K
) )
2 flfcntr.j . . . . . . . 8  |-  ( ph  ->  J  e.  Top )
3 flfcntr.c . . . . . . . . 9  |-  C  = 
U. J
43toptopon 20025 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  C ) )
52, 4sylib 201 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  C ) )
6 flfcntr.a . . . . . . 7  |-  ( ph  ->  A  C_  C )
7 resttopon 20254 . . . . . . 7  |-  ( ( J  e.  (TopOn `  C )  /\  A  C_  C )  ->  ( Jt  A )  e.  (TopOn `  A ) )
85, 6, 7syl2anc 673 . . . . . 6  |-  ( ph  ->  ( Jt  A )  e.  (TopOn `  A ) )
9 cntop2 20334 . . . . . . . 8  |-  ( F  e.  ( ( Jt  A )  Cn  K )  ->  K  e.  Top )
101, 9syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  Top )
11 flfcntr.b . . . . . . . 8  |-  B  = 
U. K
1211toptopon 20025 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  B ) )
1310, 12sylib 201 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  B ) )
14 cnflf 21095 . . . . . 6  |-  ( ( ( Jt  A )  e.  (TopOn `  A )  /\  K  e.  (TopOn `  B )
)  ->  ( F  e.  ( ( Jt  A )  Cn  K )  <->  ( F : A --> B  /\  A. a  e.  ( Fil `  A ) A. x  e.  ( ( Jt  A ) 
fLim  a ) ( F `  x )  e.  ( ( K 
fLimf  a ) `  F
) ) ) )
158, 13, 14syl2anc 673 . . . . 5  |-  ( ph  ->  ( F  e.  ( ( Jt  A )  Cn  K
)  <->  ( F : A
--> B  /\  A. a  e.  ( Fil `  A
) A. x  e.  ( ( Jt  A ) 
fLim  a ) ( F `  x )  e.  ( ( K 
fLimf  a ) `  F
) ) ) )
161, 15mpbid 215 . . . 4  |-  ( ph  ->  ( F : A --> B  /\  A. a  e.  ( Fil `  A
) A. x  e.  ( ( Jt  A ) 
fLim  a ) ( F `  x )  e.  ( ( K 
fLimf  a ) `  F
) ) )
1716simprd 470 . . 3  |-  ( ph  ->  A. a  e.  ( Fil `  A ) A. x  e.  ( ( Jt  A )  fLim  a
) ( F `  x )  e.  ( ( K  fLimf  a ) `
 F ) )
183sscls 20148 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  C )  ->  A  C_  ( ( cls `  J ) `  A
) )
192, 6, 18syl2anc 673 . . . . . 6  |-  ( ph  ->  A  C_  ( ( cls `  J ) `  A ) )
20 flfcntr.y . . . . . 6  |-  ( ph  ->  X  e.  A )
2119, 20sseldd 3419 . . . . 5  |-  ( ph  ->  X  e.  ( ( cls `  J ) `
 A ) )
226, 20sseldd 3419 . . . . . 6  |-  ( ph  ->  X  e.  C )
23 trnei 20985 . . . . . 6  |-  ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  X  e.  C )  ->  ( X  e.  ( ( cls `  J ) `  A )  <->  ( (
( nei `  J
) `  { X } )t  A )  e.  ( Fil `  A ) ) )
245, 6, 22, 23syl3anc 1292 . . . . 5  |-  ( ph  ->  ( X  e.  ( ( cls `  J
) `  A )  <->  ( ( ( nei `  J
) `  { X } )t  A )  e.  ( Fil `  A ) ) )
2521, 24mpbid 215 . . . 4  |-  ( ph  ->  ( ( ( nei `  J ) `  { X } )t  A )  e.  ( Fil `  A ) )
26 oveq2 6316 . . . . . 6  |-  ( a  =  ( ( ( nei `  J ) `
 { X }
)t 
A )  ->  (
( Jt  A )  fLim  a
)  =  ( ( Jt  A )  fLim  (
( ( nei `  J
) `  { X } )t  A ) ) )
27 oveq2 6316 . . . . . . . 8  |-  ( a  =  ( ( ( nei `  J ) `
 { X }
)t 
A )  ->  ( K  fLimf  a )  =  ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) )
2827fveq1d 5881 . . . . . . 7  |-  ( a  =  ( ( ( nei `  J ) `
 { X }
)t 
A )  ->  (
( K  fLimf  a ) `
 F )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) `  F ) )
2928eleq2d 2534 . . . . . 6  |-  ( a  =  ( ( ( nei `  J ) `
 { X }
)t 
A )  ->  (
( F `  x
)  e.  ( ( K  fLimf  a ) `  F )  <->  ( F `  x )  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) `  F ) ) )
3026, 29raleqbidv 2987 . . . . 5  |-  ( a  =  ( ( ( nei `  J ) `
 { X }
)t 
A )  ->  ( A. x  e.  (
( Jt  A )  fLim  a
) ( F `  x )  e.  ( ( K  fLimf  a ) `
 F )  <->  A. x  e.  ( ( Jt  A ) 
fLim  ( ( ( nei `  J ) `
 { X }
)t 
A ) ) ( F `  x )  e.  ( ( K 
fLimf  ( ( ( nei `  J ) `  { X } )t  A ) ) `  F ) ) )
3130adantl 473 . . . 4  |-  ( (
ph  /\  a  =  ( ( ( nei `  J ) `  { X } )t  A ) )  -> 
( A. x  e.  ( ( Jt  A ) 
fLim  a ) ( F `  x )  e.  ( ( K 
fLimf  a ) `  F
)  <->  A. x  e.  ( ( Jt  A )  fLim  (
( ( nei `  J
) `  { X } )t  A ) ) ( F `  x )  e.  ( ( K 
fLimf  ( ( ( nei `  J ) `  { X } )t  A ) ) `  F ) ) )
3225, 31rspcdv 3139 . . 3  |-  ( ph  ->  ( A. a  e.  ( Fil `  A
) A. x  e.  ( ( Jt  A ) 
fLim  a ) ( F `  x )  e.  ( ( K 
fLimf  a ) `  F
)  ->  A. x  e.  ( ( Jt  A ) 
fLim  ( ( ( nei `  J ) `
 { X }
)t 
A ) ) ( F `  x )  e.  ( ( K 
fLimf  ( ( ( nei `  J ) `  { X } )t  A ) ) `  F ) ) )
3317, 32mpd 15 . 2  |-  ( ph  ->  A. x  e.  ( ( Jt  A )  fLim  (
( ( nei `  J
) `  { X } )t  A ) ) ( F `  x )  e.  ( ( K 
fLimf  ( ( ( nei `  J ) `  { X } )t  A ) ) `  F ) )
34 neiflim 21067 . . . . 5  |-  ( ( ( Jt  A )  e.  (TopOn `  A )  /\  X  e.  A )  ->  X  e.  ( ( Jt  A ) 
fLim  ( ( nei `  ( Jt  A ) ) `  { X } ) ) )
358, 20, 34syl2anc 673 . . . 4  |-  ( ph  ->  X  e.  ( ( Jt  A )  fLim  (
( nei `  ( Jt  A ) ) `  { X } ) ) )
3620snssd 4108 . . . . . 6  |-  ( ph  ->  { X }  C_  A )
373neitr 20273 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  C  /\  { X }  C_  A )  ->  ( ( nei `  ( Jt  A ) ) `  { X } )  =  ( ( ( nei `  J ) `  { X } )t  A ) )
382, 6, 36, 37syl3anc 1292 . . . . 5  |-  ( ph  ->  ( ( nei `  ( Jt  A ) ) `  { X } )  =  ( ( ( nei `  J ) `  { X } )t  A ) )
3938oveq2d 6324 . . . 4  |-  ( ph  ->  ( ( Jt  A ) 
fLim  ( ( nei `  ( Jt  A ) ) `  { X } ) )  =  ( ( Jt  A )  fLim  ( (
( nei `  J
) `  { X } )t  A ) ) )
4035, 39eleqtrd 2551 . . 3  |-  ( ph  ->  X  e.  ( ( Jt  A )  fLim  (
( ( nei `  J
) `  { X } )t  A ) ) )
41 fveq2 5879 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4241eleq1d 2533 . . . 4  |-  ( x  =  X  ->  (
( F `  x
)  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { X } )t  A ) ) `  F )  <->  ( F `  X )  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) `  F ) ) )
4342adantl 473 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( F `  x
)  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { X } )t  A ) ) `  F )  <->  ( F `  X )  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) `  F ) ) )
4440, 43rspcdv 3139 . 2  |-  ( ph  ->  ( A. x  e.  ( ( Jt  A ) 
fLim  ( ( ( nei `  J ) `
 { X }
)t 
A ) ) ( F `  x )  e.  ( ( K 
fLimf  ( ( ( nei `  J ) `  { X } )t  A ) ) `  F )  ->  ( F `  X )  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) `  F ) ) )
4533, 44mpd 15 1  |-  ( ph  ->  ( F `  X
)  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { X } )t  A ) ) `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    C_ wss 3390   {csn 3959   U.cuni 4190   -->wf 5585   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994  TopOnctopon 19995   clsccl 20110   neicnei 20190    Cn ccn 20317   Filcfil 20938    fLim cflim 21027    fLimf cflf 21028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-fbas 19044  df-fg 19045  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033
This theorem is referenced by:  cnextfres  21162
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