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Theorem cldmreon 23102
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 22919 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
2 eqid 2737 . . . 4 𝐽 = 𝐽
32cldmre 23086 . . 3 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
41, 3syl 17 . 2 (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
5 toponuni 22920 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
65fveq2d 6910 . 2 (𝐽 ∈ (TopOn‘𝐵) → (Moore‘𝐵) = (Moore‘ 𝐽))
74, 6eleqtrrd 2844 1 (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108   cuni 4907  cfv 6561  Moorecmre 17625  Topctop 22899  TopOnctopon 22916  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-mre 17629  df-top 22900  df-topon 22917  df-cld 23027
This theorem is referenced by:  iscldtop  23103  clduni  48798
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