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Theorem cnextfres 21162
Description:  F and its extension by continuity agree on the domain of  F. (Contributed by Thierry Arnoux, 29-Aug-2020.)
Hypotheses
Ref Expression
cnextfres.c  |-  C  = 
U. J
cnextfres.b  |-  B  = 
U. K
cnextfres.j  |-  ( ph  ->  J  e.  Top )
cnextfres.k  |-  ( ph  ->  K  e.  Haus )
cnextfres.a  |-  ( ph  ->  A  C_  C )
cnextfres.1  |-  ( ph  ->  F  e.  ( ( Jt  A )  Cn  K
) )
cnextfres.x  |-  ( ph  ->  X  e.  A )
Assertion
Ref Expression
cnextfres  |-  ( ph  ->  ( ( ( JCnExt
K ) `  F
) `  X )  =  ( F `  X ) )

Proof of Theorem cnextfres
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnextfres.j . . 3  |-  ( ph  ->  J  e.  Top )
2 cnextfres.k . . 3  |-  ( ph  ->  K  e.  Haus )
3 cnextfres.1 . . . . 5  |-  ( ph  ->  F  e.  ( ( Jt  A )  Cn  K
) )
4 eqid 2471 . . . . . 6  |-  U. ( Jt  A )  =  U. ( Jt  A )
5 cnextfres.b . . . . . 6  |-  B  = 
U. K
64, 5cnf 20339 . . . . 5  |-  ( F  e.  ( ( Jt  A )  Cn  K )  ->  F : U. ( Jt  A ) --> B )
73, 6syl 17 . . . 4  |-  ( ph  ->  F : U. ( Jt  A ) --> B )
8 cnextfres.a . . . . . 6  |-  ( ph  ->  A  C_  C )
9 cnextfres.c . . . . . . 7  |-  C  = 
U. J
109restuni 20255 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  C )  ->  A  =  U. ( Jt  A ) )
111, 8, 10syl2anc 673 . . . . 5  |-  ( ph  ->  A  =  U. ( Jt  A ) )
1211feq2d 5725 . . . 4  |-  ( ph  ->  ( F : A --> B 
<->  F : U. ( Jt  A ) --> B ) )
137, 12mpbird 240 . . 3  |-  ( ph  ->  F : A --> B )
149, 5cnextfun 21157 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Haus )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  Fun  ( ( JCnExt K
) `  F )
)
151, 2, 13, 8, 14syl22anc 1293 . 2  |-  ( ph  ->  Fun  ( ( JCnExt
K ) `  F
) )
169sscls 20148 . . . . . . . 8  |-  ( ( J  e.  Top  /\  A  C_  C )  ->  A  C_  ( ( cls `  J ) `  A
) )
171, 8, 16syl2anc 673 . . . . . . 7  |-  ( ph  ->  A  C_  ( ( cls `  J ) `  A ) )
18 cnextfres.x . . . . . . 7  |-  ( ph  ->  X  e.  A )
1917, 18sseldd 3419 . . . . . 6  |-  ( ph  ->  X  e.  ( ( cls `  J ) `
 A ) )
209, 5, 1, 8, 3, 18flfcntr 21136 . . . . . 6  |-  ( ph  ->  ( F `  X
)  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { X } )t  A ) ) `  F ) )
2119, 20jca 541 . . . . 5  |-  ( ph  ->  ( X  e.  ( ( cls `  J
) `  A )  /\  ( F `  X
)  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { X } )t  A ) ) `  F ) ) )
22 sneq 3969 . . . . . . . . . 10  |-  ( x  =  X  ->  { x }  =  { X } )
2322fveq2d 5883 . . . . . . . . 9  |-  ( x  =  X  ->  (
( nei `  J
) `  { x } )  =  ( ( nei `  J
) `  { X } ) )
2423oveq1d 6323 . . . . . . . 8  |-  ( x  =  X  ->  (
( ( nei `  J
) `  { x } )t  A )  =  ( ( ( nei `  J
) `  { X } )t  A ) )
2524oveq2d 6324 . . . . . . 7  |-  ( x  =  X  ->  ( K  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
A ) )  =  ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) )
2625fveq1d 5881 . . . . . 6  |-  ( x  =  X  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { X } )t  A ) ) `  F ) )
2726opeliunxp2 4978 . . . . 5  |-  ( <. X ,  ( F `  X ) >.  e.  U_ x  e.  ( ( cls `  J ) `  A ) ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( X  e.  ( ( cls `  J
) `  A )  /\  ( F `  X
)  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { X } )t  A ) ) `  F ) ) )
2821, 27sylibr 217 . . . 4  |-  ( ph  -> 
<. X ,  ( F `
 X ) >.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
29 haustop 20424 . . . . . . 7  |-  ( K  e.  Haus  ->  K  e. 
Top )
302, 29syl 17 . . . . . 6  |-  ( ph  ->  K  e.  Top )
319, 5cnextfval 21155 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
321, 30, 13, 8, 31syl22anc 1293 . . . . 5  |-  ( ph  ->  ( ( JCnExt K
) `  F )  =  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
3332eleq2d 2534 . . . 4  |-  ( ph  ->  ( <. X ,  ( F `  X )
>.  e.  ( ( JCnExt
K ) `  F
)  <->  <. X ,  ( F `  X )
>.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
3428, 33mpbird 240 . . 3  |-  ( ph  -> 
<. X ,  ( F `
 X ) >.  e.  ( ( JCnExt K
) `  F )
)
35 df-br 4396 . . 3  |-  ( X ( ( JCnExt K
) `  F )
( F `  X
)  <->  <. X ,  ( F `  X )
>.  e.  ( ( JCnExt
K ) `  F
) )
3634, 35sylibr 217 . 2  |-  ( ph  ->  X ( ( JCnExt
K ) `  F
) ( F `  X ) )
37 funbrfv 5917 . . 3  |-  ( Fun  ( ( JCnExt K
) `  F )  ->  ( X ( ( JCnExt K ) `  F ) ( F `
 X )  -> 
( ( ( JCnExt
K ) `  F
) `  X )  =  ( F `  X ) ) )
3837imp 436 . 2  |-  ( ( Fun  ( ( JCnExt
K ) `  F
)  /\  X (
( JCnExt K ) `
 F ) ( F `  X ) )  ->  ( (
( JCnExt K ) `
 F ) `  X )  =  ( F `  X ) )
3915, 36, 38syl2anc 673 1  |-  ( ph  ->  ( ( ( JCnExt
K ) `  F
) `  X )  =  ( F `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    C_ wss 3390   {csn 3959   <.cop 3965   U.cuni 4190   U_ciun 4269   class class class wbr 4395    X. cxp 4837   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994   clsccl 20110   neicnei 20190    Cn ccn 20317   Hauscha 20401    fLimf cflf 21028  CnExtccnext 21152
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-pm 7493  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-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-cnext 21153
This theorem is referenced by:  rrhqima  28892
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