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Theorem cldss 22303
Description: A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
cldss (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝑋)

Proof of Theorem cldss
StepHypRef Expression
1 cldrcl 22300 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 22301 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simprbda 500 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
51, 4mpancom 687 1 (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cdif 3906  wss 3909   cuni 4864  cfv 6492  Topctop 22165  Clsdccld 22290
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 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6444  df-fun 6494  df-fn 6495  df-fv 6500  df-top 22166  df-cld 22293
This theorem is referenced by:  cldss2  22304  iincld  22313  uncld  22315  cldcls  22316  iuncld  22319  clsval2  22324  clsss3  22333  clsss2  22346  opncldf1  22358  restcldr  22448  lmcld  22577  nrmsep2  22630  nrmsep  22631  isnrm2  22632  regsep2  22650  cmpcld  22676  dfconn2  22693  conncompclo  22709  cldllycmp  22769  txcld  22877  ptcld  22887  imasncld  22965  kqcldsat  23007  kqnrmlem1  23017  kqnrmlem2  23018  nrmhmph  23068  ufildr  23205  metnrmlem1a  24144  metnrmlem1  24145  metnrmlem2  24146  metnrmlem3  24147  cnheiborlem  24240  cmetss  24603  bcthlem5  24615  cldssbrsiga  32560  clsun  34696  cldregopn  34699  pibt2  35784  mblfinlem3  36013  mblfinlem4  36014  ismblfin  36015  cmpfiiin  40886  kelac1  41256  stoweidlem18  44014  stoweidlem57  44053  restcls2lem  46700
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