Theorem unicls 31256

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 21635	. . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
3		sspwuni 4985	. . 3 ⊢ ((Clsd‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (Clsd‘𝐽) ⊆ 𝑋)
4	2, 3	mpbi 233	. 2 ⊢ ∪ (Clsd‘𝐽) ⊆ 𝑋
5		unicls.1	. . 3 ⊢ 𝐽 ∈ Top
6	1	topcld 21640	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
7	5, 6	ax-mp 5	. 2 ⊢ 𝑋 ∈ (Clsd‘𝐽)
8		unissel 4831	. 2 ⊢ ((∪ (Clsd‘𝐽) ⊆ 𝑋 ∧ 𝑋 ∈ (Clsd‘𝐽)) → ∪ (Clsd‘𝐽) = 𝑋)
9	4, 7, 8	mp2an 691	1 ⊢ ∪ (Clsd‘𝐽) = 𝑋

Syntax hints:   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  ‘cfv 6324  Topctop 21498  Clsdccld 21621

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-top 21499  df-cld 21624

This theorem is referenced by:  sxbrsigalem3  31640  sxbrsigalem4  31655