MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sscls Structured version   Visualization version   GIF version

Theorem sscls 20770
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 4460 . 2 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥}
2 clscld.1 . . 3 𝑋 = 𝐽
32clsval 20751 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
41, 3syl5sseqr 3633 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3555   cuni 4402   cint 4440  cfv 5847  Topctop 20617  Clsdccld 20730  clsccl 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-cld 20733  df-cls 20735
This theorem is referenced by:  iscld4  20779  elcls  20787  ntrcls0  20790  clslp  20862  restcls  20895  cncls2i  20984  nrmsep  21071  lpcls  21078  regsep2  21090  hauscmplem  21119  hauscmp  21120  clsconn  21143  conncompcld  21147  hausllycmp  21207  txcls  21317  ptclsg  21328  regr1lem  21452  kqreglem1  21454  kqreglem2  21455  kqnrmlem1  21456  kqnrmlem2  21457  fclscmpi  21743  flfcntr  21757  cnextfres  21783  clssubg  21822  tsmsid  21853  cnllycmp  22663  clsocv  22957  relcmpcmet  23023  bcthlem2  23030  bcthlem4  23032  limcnlp  23548  opnbnd  31959  opnregcld  31964  cldregopn  31965  heibor1lem  33237  heiborlem8  33246
  Copyright terms: Public domain W3C validator