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Theorem fcfval 17744
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 17653 . . . . 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 10 . . . 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 732 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  j  =  J )
43unieqd 3854 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  U. J )
5 toponuni 16681 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
65ad2antrr 706 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  X  =  U. J )
74, 6eqtr4d 2331 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  X )
8 unieq 3852 . . . . . . . 8  |-  ( f  =  L  ->  U. f  =  U. L )
98ad2antll 709 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  U. L )
10 filunibas 17592 . . . . . . . 8  |-  ( L  e.  ( Fil `  Y
)  ->  U. L  =  Y )
1110ad2antlr 707 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. L  =  Y )
129, 11eqtrd 2328 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  Y )
137, 12oveq12d 5892 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  ^m  U. f )  =  ( X  ^m  Y ) )
147oveq1d 5889 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  FilMap  g )  =  ( X  FilMap  g ) )
15 simprr 733 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  f  =  L )
1614, 15fveq12d 5547 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( ( U. j  FilMap  g ) `
 f )  =  ( ( X  FilMap  g ) `  L ) )
173, 16oveq12d 5892 . . . . 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 4114 . . . 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 16680 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2019adantr 451 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  J  e.  Top )
21 fvssunirn 5567 . . . . . 6  |-  ( Fil `  Y )  C_  U. ran  Fil
2221sseli 3189 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  U.
ran  Fil )
2322adantl 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  L  e.  U. ran  Fil )
24 ovex 5899 . . . . . 6  |-  ( X  ^m  Y )  e. 
_V
2524mptex 5762 . . . . 5  |-  ( g  e.  ( X  ^m  Y )  |->  ( J 
fClus  ( ( X  FilMap  g ) `  L ) ) )  e.  _V
2625a1i 10 . . . 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 5991 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fClusf  L )  =  ( g  e.  ( X  ^m  Y ) 
|->  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) ) )
28273adant3 975 . 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 447 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  g  =  F )
3029oveq2d 5890 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  ( X  FilMap  g )  =  ( X  FilMap  F ) )
3130fveq1d 5543 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  (
( X  FilMap  g ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
3231oveq2d 5890 . 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 16682 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
34 filtop 17566 . . . 4  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
35 elmapg 6801 . . . 4  |-  ( ( X  e.  J  /\  Y  e.  L )  ->  ( F  e.  ( X  ^m  Y )  <-> 
F : Y --> X ) )
3633, 34, 35syl2an 463 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( F  e.  ( X  ^m  Y )  <->  F : Y
--> X ) )
3736biimp3ar 1282 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F  e.  ( X  ^m  Y
) )
38 ovex 5899 . . 3  |-  ( J 
fClus  ( ( X  FilMap  F ) `  L ) )  e.  _V
3938a1i 10 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( J  fClus  ( ( X 
FilMap  F ) `  L
) )  e.  _V )
4028, 32, 37, 39fvmptd 5622 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   U.cuni 3843    e. cmpt 4093   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^m cmap 6788   Topctop 16647  TopOnctopon 16648   Filcfil 17556    FilMap cfm 17644    fClus cfcls 17647    fClusf cfcf 17648
This theorem is referenced by:  isfcf  17745  fcfelbas  17747  flfssfcf  17749  uffcfflf  17750  cnpfcfi  17751  cnpfcf  17752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-fbas 17536  df-fil 17557  df-fcf 17653
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