Theorem clscld 22198

Description: The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clscld	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))

Proof of Theorem clscld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clscld.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	clsval 22188	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
3	1	topcld 22186	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
4	3	anim1i 615	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋))
5		sseq2 3947	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋))
6	5	elrab 3624	. . . . 5 ⊢ (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋))
7	4, 6	sylibr 233	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
8	7	ne0d 4269	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ≠ ∅)
9		ssrab2 4013	. . 3 ⊢ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ (Clsd‘𝐽)
10		intcld 22191	. . 3 ⊢ (({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ (Clsd‘𝐽)) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ (Clsd‘𝐽))
11	8, 9, 10	sylancl 586	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ (Clsd‘𝐽))
12	2, 11	eqeltrd 2839	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839  ∩ cint 4879  ‘cfv 6433  Topctop 22042  Clsdccld 22167  clsccl 22169

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170  df-cls 22172

This theorem is referenced by:  clsf  22199  clsss3  22210  iscld3  22215  clsidm  22218  restcls  22332  cncls2i  22421  nrmsep  22508  lpcls  22515  regsep2  22527  hauscmplem  22557  hausllycmp  22645  txcls  22755  ptclsg  22766  regr1lem  22890  kqreglem1  22892  kqreglem2  22893  kqnrmlem1  22894  kqnrmlem2  22895  fclscmpi  23180  tgptsmscld  23302  cnllycmp  24119  clsocv  24414  cmpcmet  24483  cncmet  24486  limcnlp  25042  clsun  34517  cldregopn  34520  heibor1lem  35967  iscnrm3rlem2  46235  iscnrm3rlem5  46238