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Theorem cnextval 19638
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 4104 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
21oveq2d 6112 . . 3  |-  ( j  =  J  ->  ( U. k  ^pm  U. j
)  =  ( U. k  ^pm  U. J ) )
3 fveq2 5696 . . . . 5  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
43fveq1d 5698 . . . 4  |-  ( j  =  J  ->  (
( cls `  j
) `  dom  f )  =  ( ( cls `  J ) `  dom  f ) )
5 fveq2 5696 . . . . . . . . 9  |-  ( j  =  J  ->  ( nei `  j )  =  ( nei `  J
) )
65fveq1d 5698 . . . . . . . 8  |-  ( j  =  J  ->  (
( nei `  j
) `  { x } )  =  ( ( nei `  J
) `  { x } ) )
76oveq1d 6111 . . . . . . 7  |-  ( j  =  J  ->  (
( ( nei `  j
) `  { x } )t  dom  f )  =  ( ( ( nei `  J ) `  {
x } )t  dom  f
) )
87oveq2d 6112 . . . . . 6  |-  ( j  =  J  ->  (
k  fLimf  ( ( ( nei `  j ) `
 { x }
)t 
dom  f ) )  =  ( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) )
98fveq1d 5698 . . . . 5  |-  ( j  =  J  ->  (
( k  fLimf  ( ( ( nei `  j
) `  { x } )t  dom  f ) ) `
 f )  =  ( ( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) )
109xpeq2d 4869 . . . 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 4201 . . 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 4375 . 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 4104 . . . 4  |-  ( k  =  K  ->  U. k  =  U. K )
1413oveq1d 6111 . . 3  |-  ( k  =  K  ->  ( U. k  ^pm  U. J
)  =  ( U. K  ^pm  U. J ) )
15 oveq1 6103 . . . . . 6  |-  ( k  =  K  ->  (
k  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
dom  f ) )  =  ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) )
1615fveq1d 5698 . . . . 5  |-  ( k  =  K  ->  (
( k  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) )
1716xpeq2d 4869 . . . 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 4202 . . 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 4375 . 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 19637 . 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 6121 . . 3  |-  ( U. K  ^pm  U. J )  e.  _V
2221mptex 5953 . 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 6231 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 1369    e. wcel 1756   {csn 3882   U.cuni 4096   U_ciun 4176    e. cmpt 4355    X. cxp 4843   dom cdm 4845   ` cfv 5423  (class class class)co 6096    ^pm cpm 7220   ↾t crest 14364   Topctop 18503   clsccl 18627   neicnei 18706    fLimf cflf 19513  CnExtccnext 19636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-cnext 19637
This theorem is referenced by:  cnextfval  19639
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