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

Theorem sscls 23048
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 4966 . 2 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥}
2 clscld.1 . . 3 𝑋 = 𝐽
32clsval 23029 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
41, 3sseqtrrid 4032 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {crab 3419  wss 3946   cuni 4905   cint 4946  cfv 6546  Topctop 22883  Clsdccld 23008  clsccl 23010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-top 22884  df-cld 23011  df-cls 23013
This theorem is referenced by:  iscld4  23057  elcls  23065  ntrcls0  23068  clslp  23140  restcls  23173  cncls2i  23262  nrmsep  23349  lpcls  23356  regsep2  23368  hauscmplem  23398  hauscmp  23399  clsconn  23422  conncompcld  23426  hausllycmp  23486  txcls  23596  ptclsg  23607  regr1lem  23731  kqreglem1  23733  kqreglem2  23734  kqnrmlem1  23735  kqnrmlem2  23736  fclscmpi  24021  flfcntr  24035  cnextfres  24061  clssubg  24101  tsmsid  24132  cnllycmp  24970  clsocv  25266  relcmpcmet  25334  bcthlem2  25341  bcthlem4  25343  limcnlp  25895  opnbnd  36050  opnregcld  36055  cldregopn  36056  heibor1lem  37523  heiborlem8  37532  sepdisj  48294  iscnrm3rlem4  48313
  Copyright terms: Public domain W3C validator