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Theorem unicls 29103
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
StepHypRef Expression
1 unicls.2 . . . 4 𝑋 = 𝐽
21cldss2 20568 . . 3 (Clsd‘𝐽) ⊆ 𝒫 𝑋
3 sspwuni 4445 . . 3 ((Clsd‘𝐽) ⊆ 𝒫 𝑋 (Clsd‘𝐽) ⊆ 𝑋)
42, 3mpbi 218 . 2 (Clsd‘𝐽) ⊆ 𝑋
5 unicls.1 . . 3 𝐽 ∈ Top
61topcld 20573 . . 3 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
75, 6ax-mp 5 . 2 𝑋 ∈ (Clsd‘𝐽)
8 unissel 4302 . 2 (( (Clsd‘𝐽) ⊆ 𝑋𝑋 ∈ (Clsd‘𝐽)) → (Clsd‘𝐽) = 𝑋)
94, 7, 8mp2an 703 1 (Clsd‘𝐽) = 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1938  wss 3444  𝒫 cpw 4011   cuni 4270  cfv 5689  Topctop 20441  Clsdccld 20554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-iota 5653  df-fun 5691  df-fn 5692  df-fv 5697  df-top 20445  df-cld 20557
This theorem is referenced by:  sxbrsigalem3  29487  sxbrsigalem4  29502
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