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Theorem unicls 30279
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 21056 . . 3 (Clsd‘𝐽) ⊆ 𝒫 𝑋
3 sspwuni 4763 . . 3 ((Clsd‘𝐽) ⊆ 𝒫 𝑋 (Clsd‘𝐽) ⊆ 𝑋)
42, 3mpbi 220 . 2 (Clsd‘𝐽) ⊆ 𝑋
5 unicls.1 . . 3 𝐽 ∈ Top
61topcld 21061 . . 3 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
75, 6ax-mp 5 . 2 𝑋 ∈ (Clsd‘𝐽)
8 unissel 4620 . 2 (( (Clsd‘𝐽) ⊆ 𝑋𝑋 ∈ (Clsd‘𝐽)) → (Clsd‘𝐽) = 𝑋)
94, 7, 8mp2an 710 1 (Clsd‘𝐽) = 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  wss 3715  𝒫 cpw 4302   cuni 4588  cfv 6049  Topctop 20920  Clsdccld 21042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-top 20921  df-cld 21045
This theorem is referenced by:  sxbrsigalem3  30664  sxbrsigalem4  30679
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