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Theorem cnextval 20324
Description: The function applying continuous extension to a given function  f. (Contributed by Thierry Arnoux, 1-Dec-2017.)
Assertion
Ref Expression
cnextval  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( JCnExt K )  =  ( f  e.  ( U. K  ^pm  U. J )  |->  U_ x  e.  ( ( cls `  J
) `  dom  f ) ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) ) ) )
Distinct variable groups:    x, f, J    f, K, x

Proof of Theorem cnextval
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4253 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
21oveq2d 6300 . . 3  |-  ( j  =  J  ->  ( U. k  ^pm  U. j
)  =  ( U. k  ^pm  U. J ) )
3 fveq2 5866 . . . . 5  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
43fveq1d 5868 . . . 4  |-  ( j  =  J  ->  (
( cls `  j
) `  dom  f )  =  ( ( cls `  J ) `  dom  f ) )
5 fveq2 5866 . . . . . . . . 9  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
65fveq1d 5868 . . . . . . . 8  |-  ( j  =  J  ->  (
( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
76oveq1d 6299 . . . . . . 7  |-  ( j  =  J  ->  (
( ( nei `  j
) `  { x } )t  dom  f )  =  ( ( ( nei `  J ) `  {
x } )t  dom  f
) )
87oveq2d 6300 . . . . . 6  |-  ( j  =  J  ->  (
k  fLimf  ( ( ( nei `  j ) `
 { x }
)t 
dom  f ) )  =  ( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) )
98fveq1d 5868 . . . . 5  |-  ( j  =  J  ->  (
( k  fLimf  ( ( ( nei `  j
) `  { x } )t  dom  f ) ) `
 f )  =  ( ( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) )
109xpeq2d 5023 . . . 4  |-  ( j  =  J  ->  ( { x }  X.  ( ( k  fLimf  ( ( ( nei `  j
) `  { x } )t  dom  f ) ) `
 f ) )  =  ( { x }  X.  ( ( k 
fLimf  ( ( ( nei `  J ) `  {
x } )t  dom  f
) ) `  f
) ) )
114, 10iuneq12d 4351 . . 3  |-  ( j  =  J  ->  U_ x  e.  ( ( cls `  j
) `  dom  f ) ( { x }  X.  ( ( k  fLimf  ( ( ( nei `  j
) `  { x } )t  dom  f ) ) `
 f ) )  =  U_ x  e.  ( ( cls `  J
) `  dom  f ) ( { x }  X.  ( ( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) ) )
122, 11mpteq12dv 4525 . 2  |-  ( j  =  J  ->  (
f  e.  ( U. k  ^pm  U. j ) 
|->  U_ x  e.  ( ( cls `  j
) `  dom  f ) ( { x }  X.  ( ( k  fLimf  ( ( ( nei `  j
) `  { x } )t  dom  f ) ) `
 f ) ) )  =  ( f  e.  ( U. k  ^pm  U. J )  |->  U_ x  e.  ( ( cls `  J ) `  dom  f ) ( { x }  X.  (
( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) ) ) )
13 unieq 4253 . . . 4  |-  ( k  =  K  ->  U. k  =  U. K )
1413oveq1d 6299 . . 3  |-  ( k  =  K  ->  ( U. k  ^pm  U. J
)  =  ( U. K  ^pm  U. J ) )
15 oveq1 6291 . . . . . 6  |-  ( k  =  K  ->  (
k  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
dom  f ) )  =  ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) )
1615fveq1d 5868 . . . . 5  |-  ( k  =  K  ->  (
( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) )
1716xpeq2d 5023 . . . 4  |-  ( k  =  K  ->  ( { x }  X.  ( ( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) )  =  ( { x }  X.  ( ( K 
fLimf  ( ( ( nei `  J ) `  {
x } )t  dom  f
) ) `  f
) ) )
1817iuneq2d 4352 . . 3  |-  ( k  =  K  ->  U_ x  e.  ( ( cls `  J
) `  dom  f ) ( { x }  X.  ( ( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) )  =  U_ x  e.  ( ( cls `  J
) `  dom  f ) ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) ) )
1914, 18mpteq12dv 4525 . 2  |-  ( k  =  K  ->  (
f  e.  ( U. k  ^pm  U. J ) 
|->  U_ x  e.  ( ( cls `  J
) `  dom  f ) ( { x }  X.  ( ( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) ) )  =  ( f  e.  ( U. K  ^pm  U. J )  |->  U_ x  e.  ( ( cls `  J ) `  dom  f ) ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) ) ) )
20 df-cnext 20323 . 2  |- CnExt  =  ( j  e.  Top , 
k  e.  Top  |->  ( f  e.  ( U. k  ^pm  U. j ) 
|->  U_ x  e.  ( ( cls `  j
) `  dom  f ) ( { x }  X.  ( ( k  fLimf  ( ( ( nei `  j
) `  { x } )t  dom  f ) ) `
 f ) ) ) )
21 ovex 6309 . . 3  |-  ( U. K  ^pm  U. J )  e.  _V
2221mptex 6131 . 2  |-  ( f  e.  ( U. K  ^pm  U. J )  |->  U_ x  e.  ( ( cls `  J ) `  dom  f ) ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) ) )  e.  _V
2312, 19, 20, 22ovmpt2 6422 1  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( JCnExt K )  =  ( f  e.  ( U. K  ^pm  U. J )  |->  U_ x  e.  ( ( cls `  J
) `  dom  f ) ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4027   U.cuni 4245   U_ciun 4325    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   ` cfv 5588  (class class class)co 6284    ^pm cpm 7421   ↾t crest 14676   Topctop 19189   clsccl 19313   neicnei 19392    fLimf cflf 20199  CnExtccnext 20322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-cnext 20323
This theorem is referenced by:  cnextfval  20325
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