Theorem clduni 48889

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 22821	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2	1	biimpi 216	. 2 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽))
3		cldmreon 22997	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4		mreuni 17520	. 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 2109  ∪ cuni 4861  ‘cfv 6486  Moorecmre 17502  Topctop 22796  TopOnctopon 22813  Clsdccld 22919

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-mre 17506  df-top 22797  df-topon 22814  df-cld 22922

This theorem is referenced by:  clddisj  48892