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

Theorem clscld 21072
Description: The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clscld ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))

Proof of Theorem clscld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 clscld.1 . . 3 𝑋 = 𝐽
21clsval 21062 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
31topcld 21060 . . . . . 6 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
43anim1i 602 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋))
5 sseq2 3776 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
65elrab 3515 . . . . 5 (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋))
74, 6sylibr 224 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
87ne0d 4070 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅)
9 ssrab2 3836 . . 3 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ (Clsd‘𝐽)
10 intcld 21065 . . 3 (({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅ ∧ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ (Clsd‘𝐽)) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ (Clsd‘𝐽))
118, 9, 10sylancl 574 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ (Clsd‘𝐽))
122, 11eqeltrd 2850 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  {crab 3065  wss 3723  c0 4063   cuni 4575   cint 4612  cfv 6030  Topctop 20918  Clsdccld 21041  clsccl 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-top 20919  df-cld 21044  df-cls 21046
This theorem is referenced by:  clsf  21073  clsss3  21084  iscld3  21089  clsidm  21092  restcls  21206  cncls2i  21295  nrmsep  21382  lpcls  21389  regsep2  21401  hauscmplem  21430  hausllycmp  21518  txcls  21628  ptclsg  21639  regr1lem  21763  kqreglem1  21765  kqreglem2  21766  kqnrmlem1  21767  kqnrmlem2  21768  fclscmpi  22053  tgptsmscld  22174  cnllycmp  22975  clsocv  23268  cmpcmet  23335  cncmet  23338  limcnlp  23862  clsun  32660  cldregopn  32663  heibor1lem  33938
  Copyright terms: Public domain W3C validator