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

Theorem cldcls 21650
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 21634 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 eqid 2824 . . . 4 𝐽 = 𝐽
32cldss 21637 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
42clsval 21645 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
51, 3, 4syl2anc 587 . 2 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
6 intmin 4882 . 2 (𝑆 ∈ (Clsd‘𝐽) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} = 𝑆)
75, 6eqtrd 2859 1 (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3137  wss 3919   cuni 4824   cint 4862  cfv 6343  Topctop 21501  Clsdccld 21624  clsccl 21626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-top 21502  df-cld 21627  df-cls 21629
This theorem is referenced by:  iscld3  21672  clstop  21677  clsss2  21680  cls0  21688  cncls2  21881  lmcld  21911  fclscmp  22638  metnrmlem1a  23466  lebnumlem1  23569  cmetss  23923  minveclem4  24039  hauseqcn  31198
  Copyright terms: Public domain W3C validator