Theorem unicls 34087

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 23013	. . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
3		sspwuni 5029	. . 3 ⊢ ((Clsd‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (Clsd‘𝐽) ⊆ 𝑋)
4	2, 3	mpbi 231	. 2 ⊢ ∪ (Clsd‘𝐽) ⊆ 𝑋
5		unicls.1	. . 3 ⊢ 𝐽 ∈ Top
6	1	topcld 23018	. . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
7	5, 6	ax-mp 5	. 2 ⊢ 𝑋 ∈ (Clsd‘𝐽)
8		unissel 4870	. 2 ⊢ ((∪ (Clsd‘𝐽) ⊆ 𝑋 ∧ 𝑋 ∈ (Clsd‘𝐽)) → ∪ (Clsd‘𝐽) = 𝑋)
9	4, 7, 8	mp2an 698	1 ⊢ ∪ (Clsd‘𝐽) = 𝑋

Syntax hints:   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  𝒫 cpw 4529  ∪ cuni 4838  ‘cfv 6485  Topctop 22876  Clsdccld 22999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493  df-top 22877  df-cld 23002

This theorem is referenced by:  sxbrsigalem3  34456  sxbrsigalem4  34471