Theorem clduni 49388

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 22893	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2	1	biimpi 216	. 2 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽))
3		cldmreon 23069	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4		mreuni 17553	. 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 1542   ∈ wcel 2114  ∪ cuni 4851  ‘cfv 6492  Moorecmre 17535  Topctop 22868  TopOnctopon 22885  Clsdccld 22991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-mre 17539  df-top 22869  df-topon 22886  df-cld 22994

This theorem is referenced by:  clddisj  49391