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Theorem clscld 20974
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 20964 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
31topcld 20962 . . . . . 6 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
43anim1i 593 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋))
5 sseq2 3733 . . . . . 6 (𝑥 = 𝑋 → (𝑆𝑥𝑆𝑋))
65elrab 3469 . . . . 5 (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆𝑋))
74, 6sylibr 224 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
8 ne0i 4029 . . . 4 (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅)
97, 8syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅)
10 ssrab2 3793 . . 3 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ (Clsd‘𝐽)
11 intcld 20967 . . 3 (({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ≠ ∅ ∧ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ (Clsd‘𝐽)) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ (Clsd‘𝐽))
129, 10, 11sylancl 697 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ∈ (Clsd‘𝐽))
132, 12eqeltrd 2803 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  wne 2896  {crab 3018  wss 3680  c0 4023   cuni 4544   cint 4583  cfv 6001  Topctop 20821  Clsdccld 20943  clsccl 20945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-top 20822  df-cld 20946  df-cls 20948
This theorem is referenced by:  clsf  20975  clsss3  20986  iscld3  20991  clsidm  20994  restcls  21108  cncls2i  21197  nrmsep  21284  lpcls  21291  regsep2  21303  hauscmplem  21332  hausllycmp  21420  txcls  21530  ptclsg  21541  regr1lem  21665  kqreglem1  21667  kqreglem2  21668  kqnrmlem1  21669  kqnrmlem2  21670  fclscmpi  21955  tgptsmscld  22076  cnllycmp  22877  clsocv  23170  cmpcmet  23237  cncmet  23240  limcnlp  23762  clsun  32550  cldregopn  32553  heibor1lem  33840
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