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Theorem cnextfval 20428
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 20427 . . 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 465 . 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 461 . . . . . 6  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  f  =  F )
43dmeqd 5211 . . . . 5  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  dom  f  =  dom  F )
5 simplrl 759 . . . . . 6  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  F : A
--> B )
6 fdm 5741 . . . . . 6  |-  ( F : A --> B  ->  dom  F  =  A )
75, 6syl 16 . . . . 5  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  dom  F  =  A )
84, 7eqtrd 2508 . . . 4  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  dom  f  =  A )
98fveq2d 5876 . . 3  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  ( ( cls `  J ) `  dom  f )  =  ( ( cls `  J
) `  A )
)
108oveq2d 6311 . . . . . 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 6311 . . . . 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 5878 . . . 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 5029 . . 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 4357 . 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 6592 . . . 4  |-  ( K  e.  Top  ->  U. K  e.  _V )
1615ad2antlr 726 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  U. K  e.  _V )
17 uniexg 6592 . . . 4  |-  ( J  e.  Top  ->  U. J  e.  _V )
1817ad2antrr 725 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  U. J  e.  _V )
19 eqid 2467 . . . . . 6  |-  A  =  A
20 cnextfval.2 . . . . . 6  |-  B  = 
U. K
2119, 20feq23i 5731 . . . . 5  |-  ( F : A --> B  <->  F : A
--> U. K )
2221biimpi 194 . . . 4  |-  ( F : A --> B  ->  F : A --> U. K
)
2322ad2antrl 727 . . 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 3534 . . . . 5  |-  ( A 
C_  X  <->  A  C_  U. J
)
2625biimpi 194 . . . 4  |-  ( A 
C_  X  ->  A  C_ 
U. J )
2726ad2antll 728 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  A  C_ 
U. J )
28 elpm2r 7448 . . 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 1229 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  F  e.  ( U. K  ^pm  U. J ) )
30 fvex 5882 . . . 4  |-  ( ( cls `  J ) `
 A )  e. 
_V
31 snex 4694 . . . . 5  |-  { x }  e.  _V
32 fvex 5882 . . . . 5  |-  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F )  e.  _V
3331, 32xpex 6599 . . . 4  |-  ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  e. 
_V
3430, 33iunex 6775 . . 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 5962 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   {csn 4033   U.cuni 4251   U_ciun 4331    |-> cmpt 4511    X. cxp 5003   dom cdm 5005   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^pm cpm 7433   ↾t crest 14692   Topctop 19261   clsccl 19385   neicnei 19464    fLimf cflf 20302  CnExtccnext 20425
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-pm 7435  df-cnext 20426
This theorem is referenced by:  cnextrel  20429  cnextfun  20430  cnextfvval  20431  cnextf  20432
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