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Theorem cldcls 21179
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 21163 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 eqid 2803 . . . 4 𝐽 = 𝐽
32cldss 21166 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
42clsval 21174 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
51, 3, 4syl2anc 580 . 2 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
6 intmin 4691 . 2 (𝑆 ∈ (Clsd‘𝐽) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} = 𝑆)
75, 6eqtrd 2837 1 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  {crab 3097  wss 3773   cuni 4632   cint 4671  cfv 6105  Topctop 21030  Clsdccld 21153  clsccl 21155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-int 4672  df-iun 4716  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-top 21031  df-cld 21156  df-cls 21158
This theorem is referenced by:  iscld3  21201  clsss2  21209  cncls2  21410  lmcld  21440  fclscmp  22166  metnrmlem1a  22993  lebnumlem1  23092  cmetss  23446  minveclem4  23546  hauseqcn  30461
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