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Theorem flfcntr 21058
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 19948 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  C ) )
52, 4sylib 200 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  C ) )
6 flfcntr.a . . . . . . 7  |-  ( ph  ->  A  C_  C )
7 resttopon 20177 . . . . . . 7  |-  ( ( J  e.  (TopOn `  C )  /\  A  C_  C )  ->  ( Jt  A )  e.  (TopOn `  A ) )
85, 6, 7syl2anc 667 . . . . . 6  |-  ( ph  ->  ( Jt  A )  e.  (TopOn `  A ) )
9 cntop2 20257 . . . . . . . 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 19948 . . . . . . 7  |-  ( K  e.  Top  <->  K  e.  (TopOn `  B ) )
1310, 12sylib 200 . . . . . 6  |-  ( ph  ->  K  e.  (TopOn `  B ) )
14 cnflf 21017 . . . . . 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 667 . . . . 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 214 . . . 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 465 . . 3  |-  ( ph  ->  A. a  e.  ( Fil `  A ) A. x  e.  ( ( Jt  A )  fLim  a
) ( F `  x )  e.  ( ( K  fLimf  a ) `
 F ) )
183sscls 20071 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  C )  ->  A  C_  ( ( cls `  J ) `  A
) )
192, 6, 18syl2anc 667 . . . . . 6  |-  ( ph  ->  A  C_  ( ( cls `  J ) `  A ) )
20 flfcntr.y . . . . . 6  |-  ( ph  ->  X  e.  A )
2119, 20sseldd 3433 . . . . 5  |-  ( ph  ->  X  e.  ( ( cls `  J ) `
 A ) )
226, 20sseldd 3433 . . . . . 6  |-  ( ph  ->  X  e.  C )
23 trnei 20907 . . . . . 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 1268 . . . . 5  |-  ( ph  ->  ( X  e.  ( ( cls `  J
) `  A )  <->  ( ( ( nei `  J
) `  { X } )t  A )  e.  ( Fil `  A ) ) )
2521, 24mpbid 214 . . . 4  |-  ( ph  ->  ( ( ( nei `  J ) `  { X } )t  A )  e.  ( Fil `  A ) )
26 oveq2 6298 . . . . . 6  |-  ( a  =  ( ( ( nei `  J ) `
 { X }
)t 
A )  ->  (
( Jt  A )  fLim  a
)  =  ( ( Jt  A )  fLim  (
( ( nei `  J
) `  { X } )t  A ) ) )
27 oveq2 6298 . . . . . . . 8  |-  ( a  =  ( ( ( nei `  J ) `
 { X }
)t 
A )  ->  ( K  fLimf  a )  =  ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) )
2827fveq1d 5867 . . . . . . 7  |-  ( a  =  ( ( ( nei `  J ) `
 { X }
)t 
A )  ->  (
( K  fLimf  a ) `
 F )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) `  F ) )
2928eleq2d 2514 . . . . . 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 3001 . . . . 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 468 . . . 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 3153 . . 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 20989 . . . . 5  |-  ( ( ( Jt  A )  e.  (TopOn `  A )  /\  X  e.  A )  ->  X  e.  ( ( Jt  A ) 
fLim  ( ( nei `  ( Jt  A ) ) `  { X } ) ) )
358, 20, 34syl2anc 667 . . . 4  |-  ( ph  ->  X  e.  ( ( Jt  A )  fLim  (
( nei `  ( Jt  A ) ) `  { X } ) ) )
3620snssd 4117 . . . . . 6  |-  ( ph  ->  { X }  C_  A )
373neitr 20196 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  C  /\  { X }  C_  A )  ->  ( ( nei `  ( Jt  A ) ) `  { X } )  =  ( ( ( nei `  J ) `  { X } )t  A ) )
382, 6, 36, 37syl3anc 1268 . . . . 5  |-  ( ph  ->  ( ( nei `  ( Jt  A ) ) `  { X } )  =  ( ( ( nei `  J ) `  { X } )t  A ) )
3938oveq2d 6306 . . . 4  |-  ( ph  ->  ( ( Jt  A ) 
fLim  ( ( nei `  ( Jt  A ) ) `  { X } ) )  =  ( ( Jt  A )  fLim  ( (
( nei `  J
) `  { X } )t  A ) ) )
4035, 39eleqtrd 2531 . . 3  |-  ( ph  ->  X  e.  ( ( Jt  A )  fLim  (
( ( nei `  J
) `  { X } )t  A ) ) )
41 fveq2 5865 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4241eleq1d 2513 . . . 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 468 . . 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 3153 . 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737    C_ wss 3404   {csn 3968   U.cuni 4198   -->wf 5578   ` cfv 5582  (class class class)co 6290   ↾t crest 15319   Topctop 19917  TopOnctopon 19918   clsccl 20033   neicnei 20113    Cn ccn 20240   Filcfil 20860    fLim cflim 20949    fLimf cflf 20950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-fin 7573  df-fi 7925  df-rest 15321  df-topgen 15342  df-fbas 18967  df-fg 18968  df-top 19921  df-bases 19922  df-topon 19923  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-cn 20243  df-cnp 20244  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955
This theorem is referenced by:  cnextfres  21084
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