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Theorem clduni 45763
Description: The union of closed sets is the underlying set of the topology (the union of open sets). (Contributed by Zhi Wang, 6-Sep-2024.)
Assertion
Ref Expression
clduni (𝐽 ∈ Top → (Clsd‘𝐽) = 𝐽)

Proof of Theorem clduni
StepHypRef Expression
1 toptopon2 21681 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
21biimpi 219 . 2 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘ 𝐽))
3 cldmreon 21857 . 2 (𝐽 ∈ (TopOn‘ 𝐽) → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
4 mreuni 16986 . 2 ((Clsd‘𝐽) ∈ (Moore‘ 𝐽) → (Clsd‘𝐽) = 𝐽)
52, 3, 43syl 18 1 (𝐽 ∈ Top → (Clsd‘𝐽) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114   cuni 4806  cfv 6349  Moorecmre 16968  Topctop 21656  TopOnctopon 21673  Clsdccld 21779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-iin 4894  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-iota 6307  df-fun 6351  df-fn 6352  df-fv 6357  df-mre 16972  df-top 21657  df-topon 21674  df-cld 21782
This theorem is referenced by:  clddisj  45766
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