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Theorem clduni 48625
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 22922 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
21biimpi 216 . 2 (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘ 𝐽))
3 cldmreon 23100 . 2 (𝐽 ∈ (TopOn‘ 𝐽) → (Clsd‘𝐽) ∈ (Moore‘ 𝐽))
4 mreuni 17635 . 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 1535  wcel 2104   cuni 4915  cfv 6559  Moorecmre 17617  Topctop 22897  TopOnctopon 22914  Clsdccld 23022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7748
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4916  df-int 4955  df-iun 5001  df-iin 5002  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6511  df-fun 6561  df-fn 6562  df-fv 6567  df-mre 17621  df-top 22898  df-topon 22915  df-cld 23025
This theorem is referenced by:  clddisj  48628
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