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Theorem fcfval 19565
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 19474 . . . . 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 750 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  j  =  J )
43unieqd 4098 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  U. J )
5 toponuni 18491 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
65ad2antrr 720 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  X  =  U. J )
74, 6eqtr4d 2476 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  X )
8 unieq 4096 . . . . . . . 8  |-  ( f  =  L  ->  U. f  =  U. L )
98ad2antll 723 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  U. L )
10 filunibas 19413 . . . . . . . 8  |-  ( L  e.  ( Fil `  Y
)  ->  U. L  =  Y )
1110ad2antlr 721 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. L  =  Y )
129, 11eqtrd 2473 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  Y )
137, 12oveq12d 6108 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  ^m  U. f )  =  ( X  ^m  Y ) )
147oveq1d 6105 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  FilMap  g )  =  ( X  FilMap  g ) )
15 simprr 751 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  f  =  L )
1614, 15fveq12d 5694 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( ( U. j  FilMap  g ) `
 f )  =  ( ( X  FilMap  g ) `  L ) )
173, 16oveq12d 6108 . . . . 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 4367 . . . 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 18490 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2019adantr 462 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  J  e.  Top )
21 fvssunirn 5710 . . . . . 6  |-  ( Fil `  Y )  C_  U. ran  Fil
2221sseli 3349 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  U.
ran  Fil )
2322adantl 463 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  L  e.  U. ran  Fil )
24 ovex 6115 . . . . . 6  |-  ( X  ^m  Y )  e. 
_V
2524mptex 5945 . . . . 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 6217 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fClusf  L )  =  ( g  e.  ( X  ^m  Y ) 
|->  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) ) )
28273adant3 1003 . 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 458 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  g  =  F )
3029oveq2d 6106 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  ( X  FilMap  g )  =  ( X  FilMap  F ) )
3130fveq1d 5690 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  (
( X  FilMap  g ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
3231oveq2d 6106 . 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 18492 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
34 filtop 19387 . . . 4  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
35 elmapg 7223 . . . 4  |-  ( ( X  e.  J  /\  Y  e.  L )  ->  ( F  e.  ( X  ^m  Y )  <-> 
F : Y --> X ) )
3633, 34, 35syl2an 474 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( F  e.  ( X  ^m  Y )  <->  F : Y
--> X ) )
3736biimp3ar 1314 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F  e.  ( X  ^m  Y
) )
38 ovex 6115 . . 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 5776 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 960    = wceq 1364    e. wcel 1761   _Vcvv 2970   U.cuni 4088    e. cmpt 4347   ran crn 4837   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092    ^m cmap 7210   Topctop 18457  TopOnctopon 18458   Filcfil 19377    FilMap cfm 19465    fClus cfcls 19468    fClusf cfcf 19469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-fbas 17773  df-top 18462  df-topon 18465  df-fil 19378  df-fcf 19474
This theorem is referenced by:  isfcf  19566  fcfelbas  19568  flfssfcf  19570  uffcfflf  19571  cnpfcfi  19572  cnpfcf  19573
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