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Theorem fcfval 19606
Description: The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
fcfval  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )

Proof of Theorem fcfval
Dummy variables  f 
g  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fcf 19515 . . . . 5  |-  fClusf  =  ( j  e.  Top , 
f  e.  U. ran  Fil  |->  ( g  e.  ( U. j  ^m  U. f )  |->  ( j 
fClus  ( ( U. j  FilMap  g ) `  f
) ) ) )
21a1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  fClusf  =  ( j  e.  Top , 
f  e.  U. ran  Fil  |->  ( g  e.  ( U. j  ^m  U. f )  |->  ( j 
fClus  ( ( U. j  FilMap  g ) `  f
) ) ) ) )
3 simprl 755 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  j  =  J )
43unieqd 4101 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  U. J )
5 toponuni 18532 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
65ad2antrr 725 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  X  =  U. J )
74, 6eqtr4d 2478 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  X )
8 unieq 4099 . . . . . . . 8  |-  ( f  =  L  ->  U. f  =  U. L )
98ad2antll 728 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  U. L )
10 filunibas 19454 . . . . . . . 8  |-  ( L  e.  ( Fil `  Y
)  ->  U. L  =  Y )
1110ad2antlr 726 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. L  =  Y )
129, 11eqtrd 2475 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  Y )
137, 12oveq12d 6109 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  ^m  U. f )  =  ( X  ^m  Y ) )
147oveq1d 6106 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  FilMap  g )  =  ( X  FilMap  g ) )
15 simprr 756 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  f  =  L )
1614, 15fveq12d 5697 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( ( U. j  FilMap  g ) `
 f )  =  ( ( X  FilMap  g ) `  L ) )
173, 16oveq12d 6109 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( j  fClus  ( ( U. j  FilMap  g ) `  f
) )  =  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) )
1813, 17mpteq12dv 4370 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( g  e.  ( U. j  ^m  U. f )  |->  ( j 
fClus  ( ( U. j  FilMap  g ) `  f
) ) )  =  ( g  e.  ( X  ^m  Y ) 
|->  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) ) )
19 topontop 18531 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2019adantr 465 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  J  e.  Top )
21 fvssunirn 5713 . . . . . 6  |-  ( Fil `  Y )  C_  U. ran  Fil
2221sseli 3352 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  U.
ran  Fil )
2322adantl 466 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  L  e.  U. ran  Fil )
24 ovex 6116 . . . . . 6  |-  ( X  ^m  Y )  e. 
_V
2524mptex 5948 . . . . 5  |-  ( g  e.  ( X  ^m  Y )  |->  ( J 
fClus  ( ( X  FilMap  g ) `  L ) ) )  e.  _V
2625a1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  (
g  e.  ( X  ^m  Y )  |->  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) )  e.  _V )
272, 18, 20, 23, 26ovmpt2d 6218 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fClusf  L )  =  ( g  e.  ( X  ^m  Y ) 
|->  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) ) )
28273adant3 1008 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( J  fClusf  L )  =  ( g  e.  ( X  ^m  Y ) 
|->  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) ) )
29 simpr 461 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  g  =  F )
3029oveq2d 6107 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  ( X  FilMap  g )  =  ( X  FilMap  F ) )
3130fveq1d 5693 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  (
( X  FilMap  g ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
3231oveq2d 6107 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  ( J  fClus  ( ( X 
FilMap  g ) `  L
) )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )
33 toponmax 18533 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
34 filtop 19428 . . . 4  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
35 elmapg 7227 . . . 4  |-  ( ( X  e.  J  /\  Y  e.  L )  ->  ( F  e.  ( X  ^m  Y )  <-> 
F : Y --> X ) )
3633, 34, 35syl2an 477 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( F  e.  ( X  ^m  Y )  <->  F : Y
--> X ) )
3736biimp3ar 1319 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F  e.  ( X  ^m  Y
) )
38 ovex 6116 . . 3  |-  ( J 
fClus  ( ( X  FilMap  F ) `  L ) )  e.  _V
3938a1i 11 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( J  fClus  ( ( X 
FilMap  F ) `  L
) )  e.  _V )
4028, 32, 37, 39fvmptd 5779 1  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fClusf  L ) `
 F )  =  ( J  fClus  ( ( X  FilMap  F ) `  L ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972   U.cuni 4091    e. cmpt 4350   ran crn 4841   -->wf 5414   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093    ^m cmap 7214   Topctop 18498  TopOnctopon 18499   Filcfil 19418    FilMap cfm 19506    fClus cfcls 19509    fClusf cfcf 19510
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-fbas 17814  df-top 18503  df-topon 18506  df-fil 19419  df-fcf 19515
This theorem is referenced by:  isfcf  19607  fcfelbas  19609  flfssfcf  19611  uffcfflf  19612  cnpfcfi  19613  cnpfcf  19614
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