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Theorem cldmreon 23117
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 22934 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
2 eqid 2734 . . . 4 𝐽 = 𝐽
32cldmre 23101 . . 3 (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
41, 3syl 17 . 2 (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
5 toponuni 22935 . . 3 (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)
65fveq2d 6910 . 2 (𝐽 ∈ (TopOn‘𝐵) → (Moore‘𝐵) = (Moore‘ 𝐽))
74, 6eleqtrrd 2841 1 (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105   cuni 4911  cfv 6562  Moorecmre 17626  Topctop 22914  TopOnctopon 22931  Clsdccld 23039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-mre 17630  df-top 22915  df-topon 22932  df-cld 23042
This theorem is referenced by:  iscldtop  23118  clduni  48696
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