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Theorem cldmreon 17082
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 16915 . . 3  |-  ( J  e.  (TopOn `  B
)  ->  J  e.  Top )
2 eqid 2388 . . . 4  |-  U. J  =  U. J
32cldmre 17066 . . 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 16916 . . 3  |-  ( J  e.  (TopOn `  B
)  ->  B  =  U. J )
65fveq2d 5673 . 2  |-  ( J  e.  (TopOn `  B
)  ->  (Moore `  B
)  =  (Moore `  U. J ) )
74, 6eleqtrrd 2465 1  |-  ( J  e.  (TopOn `  B
)  ->  ( Clsd `  J )  e.  (Moore `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   U.cuni 3958   ` cfv 5395  Moorecmre 13735   Topctop 16882  TopOnctopon 16883   Clsdccld 17004
This theorem is referenced by:  iscldtop  17083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403  df-mre 13739  df-top 16887  df-topon 16890  df-cld 17007
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