Theorem clscld 23278

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 23268	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
3	1	topcld 23266	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
4	3	anim1i 627	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋))
5		sseq2 3960	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋))
6	5	elrab 3648	. . . . 5 ⊢ (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋))
7	4, 6	sylibr 237	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
8	7	ne0d 4291	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ≠ ∅)
9		ssrab2 4031	. . 3 ⊢ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ (Clsd‘𝐽)
10		intcld 23271	. . 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 598	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ (Clsd‘𝐽))
12	2, 11	eqeltrd 2862	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  {crab 3414   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  ∩ cint 4910  ‘cfv 6537  Topctop 23124  Clsdccld 23247  clsccl 23249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23125  df-cld 23250  df-cls 23252

This theorem is used by:  clsf  23279  clsss3  23290  iscld3  23295  clsidm  23298  restcls  23412  cncls2i  23501  nrmsep  23588  lpcls  23595  regsep2  23607  hauscmplem  23637  hausllycmp  23726  txcls  23836  ptclsg  23847  regr1lem  23971  kqreglem1  23973  kqreglem2  23974  kqnrmlem1  23975  kqnrmlem2  23976  fclscmpi  24261  tgptsmscld  24383  cnllycmp  25190  clsocv  25484  cmpcmet  25553  cncmet  25556  limcnlp  26112  clsun  36955  cldregopn  36958  heibor1lem  38567  iscnrm3rlem2  49875  iscnrm3rlem5  49878