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Theorem sscls 23080
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 4971 . 2 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥}
2 clscld.1 . . 3 𝑋 = 𝐽
32clsval 23061 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
41, 3sseqtrrid 4049 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  wss 3963   cuni 4912   cint 4951  cfv 6563  Topctop 22915  Clsdccld 23040  clsccl 23042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-cls 23045
This theorem is referenced by:  iscld4  23089  elcls  23097  ntrcls0  23100  clslp  23172  restcls  23205  cncls2i  23294  nrmsep  23381  lpcls  23388  regsep2  23400  hauscmplem  23430  hauscmp  23431  clsconn  23454  conncompcld  23458  hausllycmp  23518  txcls  23628  ptclsg  23639  regr1lem  23763  kqreglem1  23765  kqreglem2  23766  kqnrmlem1  23767  kqnrmlem2  23768  fclscmpi  24053  flfcntr  24067  cnextfres  24093  clssubg  24133  tsmsid  24164  cnllycmp  25002  clsocv  25298  relcmpcmet  25366  bcthlem2  25373  bcthlem4  25375  limcnlp  25928  opnbnd  36308  opnregcld  36313  cldregopn  36314  heibor1lem  37796  heiborlem8  37805  sepdisj  48721  iscnrm3rlem4  48740
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