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Theorem fcfval 20659
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 20568 . . . . 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 756 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  j  =  J )
43unieqd 4261 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  U. J )
5 toponuni 19554 . . . . . . . 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 2501 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. j  =  X )
8 unieq 4259 . . . . . . . 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 20507 . . . . . . . 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 2498 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  U. f  =  Y )
137, 12oveq12d 6314 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  ^m  U. f )  =  ( X  ^m  Y ) )
147oveq1d 6311 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( U. j  FilMap  g )  =  ( X  FilMap  g ) )
15 simprr 757 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  f  =  L )
1614, 15fveq12d 5878 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  /\  (
j  =  J  /\  f  =  L )
)  ->  ( ( U. j  FilMap  g ) `
 f )  =  ( ( X  FilMap  g ) `  L ) )
173, 16oveq12d 6314 . . . . 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 4535 . . . 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 19553 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2019adantr 465 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  J  e.  Top )
21 fvssunirn 5895 . . . . . 6  |-  ( Fil `  Y )  C_  U. ran  Fil
2221sseli 3495 . . . . 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 6324 . . . . . 6  |-  ( X  ^m  Y )  e. 
_V
2524mptex 6144 . . . . 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 6429 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
) )  ->  ( J  fClusf  L )  =  ( g  e.  ( X  ^m  Y ) 
|->  ( J  fClus  ( ( X  FilMap  g ) `  L ) ) ) )
28273adant3 1016 . 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 6312 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  ( X  FilMap  g )  =  ( X  FilMap  F ) )
3130fveq1d 5874 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  g  =  F )  ->  (
( X  FilMap  g ) `
 L )  =  ( ( X  FilMap  F ) `  L ) )
3231oveq2d 6312 . 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 19555 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
34 filtop 20481 . . . 4  |-  ( L  e.  ( Fil `  Y
)  ->  Y  e.  L )
35 elmapg 7451 . . . 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 1329 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F  e.  ( X  ^m  Y
) )
38 ovex 6324 . . 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 5961 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   U.cuni 4251    |-> cmpt 4515   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298    ^m cmap 7438   Topctop 19520  TopOnctopon 19521   Filcfil 20471    FilMap cfm 20559    fClus cfcls 20562    fClusf cfcf 20563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-fbas 18542  df-top 19525  df-topon 19528  df-fil 20472  df-fcf 20568
This theorem is referenced by:  isfcf  20660  fcfelbas  20662  flfssfcf  20664  uffcfflf  20665  cnpfcfi  20666  cnpfcf  20667
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