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Theorem cnextfval 19639
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 19638 . . 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 5047 . . . . 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 5568 . . . . . 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 2475 . . . 4  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  dom  f  =  A )
98fveq2d 5700 . . 3  |-  ( ( ( ( J  e. 
Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) )  /\  f  =  F )  ->  ( ( cls `  J ) `  dom  f )  =  ( ( cls `  J
) `  A )
)
108oveq2d 6112 . . . . . 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 6112 . . . . 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 5702 . . . 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 4869 . . 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 4201 . 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 6382 . . . 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 6382 . . . 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 2443 . . . . . 6  |-  A  =  A
20 cnextfval.2 . . . . . 6  |-  B  = 
U. K
2119, 20feq23i 5558 . . . . 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 3386 . . . . 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 7235 . . 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 1219 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X
) )  ->  F  e.  ( U. K  ^pm  U. J ) )
30 fvex 5706 . . . 4  |-  ( ( cls `  J ) `
 A )  e. 
_V
31 snex 4538 . . . . 5  |-  { x }  e.  _V
32 fvex 5706 . . . . 5  |-  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F )  e.  _V
3331, 32xpex 6513 . . . 4  |-  ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  e. 
_V
3430, 33iunex 6562 . . 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 5784 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 1369    e. wcel 1756   _Vcvv 2977    C_ wss 3333   {csn 3882   U.cuni 4096   U_ciun 4176    e. cmpt 4355    X. cxp 4843   dom cdm 4845   -->wf 5419   ` 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-8 1758  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-pow 4475  ax-pr 4536  ax-un 6377
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-pw 3867  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-pm 7222  df-cnext 19637
This theorem is referenced by:  cnextrel  19640  cnextfun  19641  cnextfvval  19642  cnextf  19643
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