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Theorem sscls 22542
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 4969 . 2 𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥}
2 clscld.1 . . 3 𝑋 = 𝐽
32clsval 22523 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
41, 3sseqtrrid 4034 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3433  wss 3947   cuni 4907   cint 4949  cfv 6540  Topctop 22377  Clsdccld 22502  clsccl 22504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 22378  df-cld 22505  df-cls 22507
This theorem is referenced by:  iscld4  22551  elcls  22559  ntrcls0  22562  clslp  22634  restcls  22667  cncls2i  22756  nrmsep  22843  lpcls  22850  regsep2  22862  hauscmplem  22892  hauscmp  22893  clsconn  22916  conncompcld  22920  hausllycmp  22980  txcls  23090  ptclsg  23101  regr1lem  23225  kqreglem1  23227  kqreglem2  23228  kqnrmlem1  23229  kqnrmlem2  23230  fclscmpi  23515  flfcntr  23529  cnextfres  23555  clssubg  23595  tsmsid  23626  cnllycmp  24454  clsocv  24749  relcmpcmet  24817  bcthlem2  24824  bcthlem4  24826  limcnlp  25377  opnbnd  35148  opnregcld  35153  cldregopn  35154  heibor1lem  36615  heiborlem8  36624  sepdisj  47459  iscnrm3rlem4  47478
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