Theorem unicls 33916

Description: The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.)

Hypotheses

Ref	Expression
unicls.1	⊢ 𝐽 ∈ Top
unicls.2	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
unicls	⊢ ∪ (Clsd‘𝐽) = 𝑋

Proof of Theorem unicls

Step	Hyp	Ref	Expression
1		unicls.2	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	cldss2 22945	. . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
3		sspwuni 5046	. . 3 ⊢ ((Clsd‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (Clsd‘𝐽) ⊆ 𝑋)
4	2, 3	mpbi 230	. 2 ⊢ ∪ (Clsd‘𝐽) ⊆ 𝑋
5		unicls.1	. . 3 ⊢ 𝐽 ∈ Top
6	1	topcld 22950	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
7	5, 6	ax-mp 5	. 2 ⊢ 𝑋 ∈ (Clsd‘𝐽)
8		unissel 4888	. 2 ⊢ ((∪ (Clsd‘𝐽) ⊆ 𝑋 ∧ 𝑋 ∈ (Clsd‘𝐽)) → ∪ (Clsd‘𝐽) = 𝑋)
9	4, 7, 8	mp2an 692	1 ⊢ ∪ (Clsd‘𝐽) = 𝑋

Syntax hints:   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856  ‘cfv 6481  Topctop 22808  Clsdccld 22931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-top 22809  df-cld 22934

This theorem is referenced by:  sxbrsigalem3  34285  sxbrsigalem4  34300