Theorem unicls 29277

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 20644	. . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
3		sspwuni 4547	. . 3 ⊢ ((Clsd‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (Clsd‘𝐽) ⊆ 𝑋)
4	2, 3	mpbi 219	. 2 ⊢ ∪ (Clsd‘𝐽) ⊆ 𝑋
5		unicls.1	. . 3 ⊢ 𝐽 ∈ Top
6	1	topcld 20649	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
7	5, 6	ax-mp 5	. 2 ⊢ 𝑋 ∈ (Clsd‘𝐽)
8		unissel 4404	. 2 ⊢ ((∪ (Clsd‘𝐽) ⊆ 𝑋 ∧ 𝑋 ∈ (Clsd‘𝐽)) → ∪ (Clsd‘𝐽) = 𝑋)
9	4, 7, 8	mp2an 704	1 ⊢ ∪ (Clsd‘𝐽) = 𝑋

Syntax hints:   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-top 20521  df-cld 20633

This theorem is referenced by:  sxbrsigalem3  29661  sxbrsigalem4  29676