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Theorem clscld 20761
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 20751 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
31topcld 20749 . . . . . 6 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
43anim1i 591 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋))
5 sseq2 3606 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
65elrab 3346 . . . . 5 (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋))
74, 6sylibr 224 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
8 ne0i 3897 . . . 4 (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅)
97, 8syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅)
10 ssrab2 3666 . . 3 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ (Clsd‘𝐽)
11 intcld 20754 . . 3 (({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅ ∧ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ (Clsd‘𝐽)) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ (Clsd‘𝐽))
129, 10, 11sylancl 693 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ (Clsd‘𝐽))
132, 12eqeltrd 2698 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  {crab 2911  wss 3555  c0 3891   cuni 4402   cint 4440  cfv 5847  Topctop 20617  Clsdccld 20730  clsccl 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-cld 20733  df-cls 20735
This theorem is referenced by:  clsf  20762  clsss3  20773  iscld3  20778  clsidm  20781  restcls  20895  cncls2i  20984  nrmsep  21071  lpcls  21078  regsep2  21090  hauscmplem  21119  hausllycmp  21207  txcls  21317  ptclsg  21328  regr1lem  21452  kqreglem1  21454  kqreglem2  21455  kqnrmlem1  21456  kqnrmlem2  21457  fclscmpi  21743  tgptsmscld  21864  cnllycmp  22663  clsocv  22957  cmpcmet  23024  cncmet  23027  limcnlp  23548  clsun  31965  cldregopn  31968  heibor1lem  33240
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