MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cldcls Structured version   Visualization version   GIF version

Theorem cldcls 22409
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 22393 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 eqid 2737 . . . 4 𝐽 = 𝐽
32cldss 22396 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
42clsval 22404 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
51, 3, 4syl2anc 585 . 2 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
6 intmin 4934 . 2 (𝑆 ∈ (Clsd‘𝐽) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} = 𝑆)
75, 6eqtrd 2777 1 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3410  wss 3915   cuni 4870   cint 4912  cfv 6501  Topctop 22258  Clsdccld 22383  clsccl 22385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-top 22259  df-cld 22386  df-cls 22388
This theorem is referenced by:  iscld3  22431  clstop  22436  clsss2  22439  cls0  22447  cncls2  22640  lmcld  22670  fclscmp  23397  metnrmlem1a  24237  lebnumlem1  24340  cmetss  24696  minveclem4  24812  hauseqcn  32519  restcls2  47020
  Copyright terms: Public domain W3C validator