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Theorem clscld 21229
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 21219 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
31topcld 21217 . . . . . 6 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
43anim1i 608 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋))
5 sseq2 3852 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
65elrab 3585 . . . . 5 (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋))
74, 6sylibr 226 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
87ne0d 4153 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅)
9 ssrab2 3914 . . 3 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ (Clsd‘𝐽)
10 intcld 21222 . . 3 (({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅ ∧ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ (Clsd‘𝐽)) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ (Clsd‘𝐽))
118, 9, 10sylancl 580 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ (Clsd‘𝐽))
122, 11eqeltrd 2906 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wne 2999  {crab 3121  wss 3798  c0 4146   cuni 4660   cint 4699  cfv 6127  Topctop 21075  Clsdccld 21198  clsccl 21200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-top 21076  df-cld 21201  df-cls 21203
This theorem is referenced by:  clsf  21230  clsss3  21241  iscld3  21246  clsidm  21249  restcls  21363  cncls2i  21452  nrmsep  21539  lpcls  21546  regsep2  21558  hauscmplem  21587  hausllycmp  21675  txcls  21785  ptclsg  21796  regr1lem  21920  kqreglem1  21922  kqreglem2  21923  kqnrmlem1  21924  kqnrmlem2  21925  fclscmpi  22210  tgptsmscld  22331  cnllycmp  23132  clsocv  23425  cmpcmet  23494  cncmet  23497  limcnlp  24048  clsun  32856  cldregopn  32859  heibor1lem  34145
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