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Theorem cldmreon 21799
 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 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))

Proof of Theorem cldmreon
StepHypRef Expression
1 topontop 21618 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
2 eqid 2758 . . . 4 𝐽 = 𝐽
32cldmre 21783 . . 3 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
41, 3syl 17 . 2 (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
5 toponuni 21619 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
65fveq2d 6666 . 2 (𝐽 ∈ (TopOn‘𝐵) → (Moore‘𝐵) = (Moore‘ 𝐽))
74, 6eleqtrrd 2855 1 (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  ∪ cuni 4801  ‘cfv 6339  Moorecmre 16916  Topctop 21598  TopOnctopon 21615  Clsdccld 21721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6298  df-fun 6341  df-fn 6342  df-fv 6347  df-mre 16920  df-top 21599  df-topon 21616  df-cld 21724 This theorem is referenced by:  iscldtop  21800  clduni  45613
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