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Theorem cldcls 21258
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 21242 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 eqid 2778 . . . 4 𝐽 = 𝐽
32cldss 21245 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
42clsval 21253 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
51, 3, 4syl2anc 579 . 2 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
6 intmin 4732 . 2 (𝑆 ∈ (Clsd‘𝐽) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} = 𝑆)
75, 6eqtrd 2814 1 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  {crab 3094  wss 3792   cuni 4673   cint 4712  cfv 6137  Topctop 21109  Clsdccld 21232  clsccl 21234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-top 21110  df-cld 21235  df-cls 21237
This theorem is referenced by:  iscld3  21280  clstop  21285  clsss2  21288  cls0  21296  cncls2  21489  lmcld  21519  fclscmp  22246  metnrmlem1a  23073  lebnumlem1  23172  cmetss  23526  minveclem4  23642  hauseqcn  30543
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