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Theorem sscls 21062
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 4647 . 2 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥}
2 clscld.1 . . 3 𝑋 = 𝐽
32clsval 21043 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
41, 3syl5sseqr 3795 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {crab 3054  wss 3715   cuni 4588   cint 4627  cfv 6049  Topctop 20900  Clsdccld 21022  clsccl 21024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-cls 21027
This theorem is referenced by:  iscld4  21071  elcls  21079  ntrcls0  21082  clslp  21154  restcls  21187  cncls2i  21276  nrmsep  21363  lpcls  21370  regsep2  21382  hauscmplem  21411  hauscmp  21412  clsconn  21435  conncompcld  21439  hausllycmp  21499  txcls  21609  ptclsg  21620  regr1lem  21744  kqreglem1  21746  kqreglem2  21747  kqnrmlem1  21748  kqnrmlem2  21749  fclscmpi  22034  flfcntr  22048  cnextfres  22074  clssubg  22113  tsmsid  22144  cnllycmp  22956  clsocv  23249  relcmpcmet  23315  bcthlem2  23322  bcthlem4  23324  limcnlp  23841  opnbnd  32626  opnregcld  32631  cldregopn  32632  heibor1lem  33921  heiborlem8  33930
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