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Theorem cldmreon 17150
Description: The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Assertion
Ref Expression
cldmreon  |-  ( J  e.  (TopOn `  B
)  ->  ( Clsd `  J )  e.  (Moore `  B ) )

Proof of Theorem cldmreon
StepHypRef Expression
1 topontop 16983 . . 3  |-  ( J  e.  (TopOn `  B
)  ->  J  e.  Top )
2 eqid 2435 . . . 4  |-  U. J  =  U. J
32cldmre 17134 . . 3  |-  ( J  e.  Top  ->  ( Clsd `  J )  e.  (Moore `  U. J ) )
41, 3syl 16 . 2  |-  ( J  e.  (TopOn `  B
)  ->  ( Clsd `  J )  e.  (Moore `  U. J ) )
5 toponuni 16984 . . 3  |-  ( J  e.  (TopOn `  B
)  ->  B  =  U. J )
65fveq2d 5724 . 2  |-  ( J  e.  (TopOn `  B
)  ->  (Moore `  B
)  =  (Moore `  U. J ) )
74, 6eleqtrrd 2512 1  |-  ( J  e.  (TopOn `  B
)  ->  ( Clsd `  J )  e.  (Moore `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   U.cuni 4007   ` cfv 5446  Moorecmre 13799   Topctop 16950  TopOnctopon 16951   Clsdccld 17072
This theorem is referenced by:  iscldtop  17151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-mre 13803  df-top 16955  df-topon 16958  df-cld 17075
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