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Theorem cldcls 20756
Description: A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)
Assertion
Ref Expression
cldcls (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)

Proof of Theorem cldcls
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cldrcl 20740 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 eqid 2621 . . . 4 𝐽 = 𝐽
32cldss 20743 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
42clsval 20751 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
51, 3, 4syl2anc 692 . 2 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
6 intmin 4462 . 2 (𝑆 ∈ (Clsd‘𝐽) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} = 𝑆)
75, 6eqtrd 2655 1 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2911  wss 3555   cuni 4402   cint 4440  cfv 5847  Topctop 20617  Clsdccld 20730  clsccl 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-cld 20733  df-cls 20735
This theorem is referenced by:  iscld3  20778  clsss2  20786  cncls2  20987  lmcld  21017  fclscmp  21744  metnrmlem1a  22569  lebnumlem1  22668  cmetss  23021  minveclem4  23111  hauseqcn  29723
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