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Theorem cnextfval 21125
Description: The continuous extension of a given function  F. (Contributed by Thierry Arnoux, 1-Dec-2017.)
Hypotheses
Ref Expression
cnextfval.1  |-  X  = 
U. J
cnextfval.2  |-  B  = 
U. K
Assertion
Ref Expression
cnextfval  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
Distinct variable groups:    x, J    x, K    x, A    x, B    x, F    x, X

Proof of Theorem cnextfval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 cnextval 21124 . . 3  |-  ( ( 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 ) ) ) )
21adantr 471 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  ( 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 ) ) ) )
3 simpr 467 . . . . . 6  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  f  =  F )
43dmeqd 5055 . . . . 5  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  dom  f  =  dom  F )
5 simplrl 775 . . . . . 6  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  F : A
--> B )
6 fdm 5755 . . . . . 6  |-  ( F : A --> B  ->  dom  F  =  A )
75, 6syl 17 . . . . 5  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  dom  F  =  A )
84, 7eqtrd 2495 . . . 4  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  dom  f  =  A )
98fveq2d 5891 . . 3  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  ( ( cls `  J ) `  dom  f )  =  ( ( cls `  J
) `  A )
)
108oveq2d 6330 . . . . . 6  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  ( (
( nei `  J
) `  { x } )t  dom  f )  =  ( ( ( nei `  J ) `  {
x } )t  A ) )
1110oveq2d 6330 . . . . 5  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  ( K  fLimf  ( ( ( nei `  J ) `  {
x } )t  dom  f
) )  =  ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) )
1211, 3fveq12d 5893 . . . 4  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  ( ( K  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
dom  f ) ) `
 f )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
1312xpeq2d 4876 . . 3  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  ( {
x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) )  =  ( { x }  X.  ( ( K 
fLimf  ( ( ( nei `  J ) `  {
x } )t  A ) ) `  F ) ) )
149, 13iuneq12d 4317 . 2  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  U_ x  e.  ( ( cls `  J
) `  dom  f ) ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  dom  f ) ) `
 f ) )  =  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
15 uniexg 6614 . . . 4  |-  ( K  e.  Top  ->  U. K  e.  _V )
1615ad2antlr 738 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  U. K  e.  _V )
17 uniexg 6614 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
1817ad2antrr 737 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  U. J  e.  _V )
19 eqid 2461 . . . . . 6  |-  A  =  A
20 cnextfval.2 . . . . . 6  |-  B  = 
U. K
2119, 20feq23i 5744 . . . . 5  |-  ( F : A --> B  <->  F : A
--> U. K )
2221biimpi 199 . . . 4  |-  ( F : A --> B  ->  F : A --> U. K
)
2322ad2antrl 739 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  F : A --> U. K )
24 cnextfval.1 . . . . . 6  |-  X  = 
U. J
2524sseq2i 3468 . . . . 5  |-  ( A 
C_  X  <->  A  C_  U. J
)
2625biimpi 199 . . . 4  |-  ( A 
C_  X  ->  A  C_ 
U. J )
2726ad2antll 740 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  A  C_ 
U. J )
28 elpm2r 7514 . . 3  |-  ( ( ( U. K  e. 
_V  /\  U. J  e. 
_V )  /\  ( F : A --> U. K  /\  A  C_  U. J
) )  ->  F  e.  ( U. K  ^pm  U. J ) )
2916, 18, 23, 27, 28syl22anc 1277 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  F  e.  ( U. K  ^pm  U. J ) )
30 fvex 5897 . . . 4  |-  ( ( cls `  J ) `
 A )  e. 
_V
31 snex 4654 . . . . 5  |-  { x }  e.  _V
32 fvex 5897 . . . . 5  |-  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F )  e.  _V
3331, 32xpex 6621 . . . 4  |-  ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  e. 
_V
3430, 33iunex 6799 . . 3  |-  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  e. 
_V
3534a1i 11 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  e. 
_V )
362, 14, 29, 35fvmptd 5976 1  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   _Vcvv 3056    C_ wss 3415   {csn 3979   U.cuni 4211   U_ciun 4291    |-> cmpt 4474    X. cxp 4850   dom cdm 4852   -->wf 5596   ` cfv 5600  (class class class)co 6314    ^pm cpm 7498   ↾t crest 15367   Topctop 19965   clsccl 20081   neicnei 20161    fLimf cflf 20998  CnExtccnext 21122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-pm 7500  df-cnext 21123
This theorem is referenced by:  cnextrel  21126  cnextfun  21127  cnextfvval  21128  cnextf  21129  cnextfres  21132
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