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Theorem sscls 22559
Description: A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
sscls ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem sscls
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssintub 4970 . 2 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥}
2 clscld.1 . . 3 𝑋 = 𝐽
32clsval 22540 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
41, 3sseqtrrid 4035 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3432  wss 3948   cuni 4908   cint 4950  cfv 6543  Topctop 22394  Clsdccld 22519  clsccl 22521
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22395  df-cld 22522  df-cls 22524
This theorem is referenced by:  iscld4  22568  elcls  22576  ntrcls0  22579  clslp  22651  restcls  22684  cncls2i  22773  nrmsep  22860  lpcls  22867  regsep2  22879  hauscmplem  22909  hauscmp  22910  clsconn  22933  conncompcld  22937  hausllycmp  22997  txcls  23107  ptclsg  23118  regr1lem  23242  kqreglem1  23244  kqreglem2  23245  kqnrmlem1  23246  kqnrmlem2  23247  fclscmpi  23532  flfcntr  23546  cnextfres  23572  clssubg  23612  tsmsid  23643  cnllycmp  24471  clsocv  24766  relcmpcmet  24834  bcthlem2  24841  bcthlem4  24843  limcnlp  25394  opnbnd  35205  opnregcld  35210  cldregopn  35211  heibor1lem  36672  heiborlem8  36681  sepdisj  47547  iscnrm3rlem4  47566
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