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Theorem fcfval 21034
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 20943 . . . . 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 762 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  j  =  J )
43unieqd 4226 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  U. J )
5 toponuni 19928 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
65ad2antrr 730 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  X  =  U. J )
74, 6eqtr4d 2466 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  X )
8 unieq 4224 . . . . . . . 8  |-  ( f  =  L  ->  U. f  =  U. L )
98ad2antll 733 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  U. L )
10 filunibas 20882 . . . . . . . 8  |-  ( L  e.  ( Fil `  Y
)  ->  U. L  =  Y )
1110ad2antlr 731 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. L  =  Y )
129, 11eqtrd 2463 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  Y )
137, 12oveq12d 6319 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  ^m  U. f )  =  ( X  ^m  Y ) )
147oveq1d 6316 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  FilMap  g )  =  ( X  FilMap  g ) )
15 simprr 764 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  f  =  L )
1614, 15fveq12d 5883 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( ( U. j  FilMap  g ) `
 f )  =  ( ( X  FilMap  g ) `  L ) )
173, 16oveq12d 6319 . . . . 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 4499 . . . 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 19927 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2019adantr 466 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  J  e.  Top )
21 fvssunirn 5900 . . . . . 6  |-  ( Fil `  Y )  C_  U. ran  Fil
2221sseli 3460 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  U.
ran  Fil )
2322adantl 467 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  L  e.  U. ran  Fil )
24 ovex 6329 . . . . . 6  |-  ( X  ^m  Y )  e. 
_V
2524mptex 6147 . . . . 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 6434 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fClusf  L )  =  ( g  e.  ( X  ^m  Y ) 
|->  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) ) )
28273adant3 1025 . 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 462 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  g  =  F )
3029oveq2d 6317 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  ( X  FilMap  g )  =  ( X  FilMap  F ) )
3130fveq1d 5879 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  (
( X  FilMap  g ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
3231oveq2d 6317 . 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 19929 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
34 filtop 20856 . . . 4  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
35 elmapg 7489 . . . 4  |-  ( ( X  e.  J  /\  Y  e.  L )  ->  ( F  e.  ( X  ^m  Y )  <-> 
F : Y --> X ) )
3633, 34, 35syl2an 479 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( F  e.  ( X  ^m  Y )  <->  F : Y
--> X ) )
3736biimp3ar 1365 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F  e.  ( X  ^m  Y
) )
38 ovex 6329 . . 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 5966 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   _Vcvv 3081   U.cuni 4216    |-> cmpt 4479   ran crn 4850   -->wf 5593   ` cfv 5597  (class class class)co 6301    |-> cmpt2 6303    ^m cmap 7476   Topctop 19903  TopOnctopon 19904   Filcfil 20846    FilMap cfm 20934    fClus cfcls 20937    fClusf cfcf 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-fbas 18954  df-top 19907  df-topon 19909  df-fil 20847  df-fcf 20943
This theorem is referenced by:  isfcf  21035  fcfelbas  21037  flfssfcf  21039  uffcfflf  21040  cnpfcfi  21041  cnpfcf  21042
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