Theorem clduni 48798

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

Step	Hyp	Ref	Expression
1		toptopon2 22924	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2	1	biimpi 216	. 2 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽))
3		cldmreon 23102	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4		mreuni 17643	. 2 ⊢ ((Clsd‘𝐽) ∈ (Moore‘∪ 𝐽) → ∪ (Clsd‘𝐽) = ∪ 𝐽)
5	2, 3, 4	3syl 18	1 ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)

Syntax hints:   → wi 4   = wceq 1540   ∈ 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:  clddisj  48801