Theorem clduni 49679

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 23075	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
2	1	biimpi 219	. 2 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽))
3		cldmreon 23251	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4		mreuni 17647	. 2 ⊢ ((Clsd‘𝐽) ∈ (Moore‘∪ 𝐽) → ∪ (Clsd‘𝐽) = ∪ 𝐽)
5	2, 3, 4	3syl 19	1 ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∪ cuni 4872  ‘cfv 6536  Moorecmre 17629  Topctop 23050  TopOnctopon 23067  Clsdccld 23173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-mre 17633  df-top 23051  df-topon 23068  df-cld 23176

This theorem is referenced by:  clddisj  49682