Theorem sscls 23021

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 ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem sscls

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssintub 4908	. 2 ⊢ 𝑆 ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}
2		clscld.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	2	clsval 23002	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
4	1, 3	sseqtrrid 3965	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389   ⊆ wss 3889  ∪ cuni 4850  ∩ cint 4889  ‘cfv 6498  Topctop 22858  Clsdccld 22981  clsccl 22983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-top 22859  df-cld 22984  df-cls 22986

This theorem is referenced by:  iscld4  23030  elcls  23038  ntrcls0  23041  clslp  23113  restcls  23146  cncls2i  23235  nrmsep  23322  lpcls  23329  regsep2  23341  hauscmplem  23371  hauscmp  23372  clsconn  23395  conncompcld  23399  hausllycmp  23459  txcls  23569  ptclsg  23580  regr1lem  23704  kqreglem1  23706  kqreglem2  23707  kqnrmlem1  23708  kqnrmlem2  23709  fclscmpi  23994  flfcntr  24008  cnextfres  24034  clssubg  24074  tsmsid  24105  cnllycmp  24923  clsocv  25217  relcmpcmet  25285  bcthlem2  25292  bcthlem4  25294  limcnlp  25845  opnbnd  36507  opnregcld  36512  cldregopn  36513  heibor1lem  38130  heiborlem8  38139  sepdisj  49400  iscnrm3rlem4  49418