Theorem clscld 23395

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 23385	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
3	1	topcld 23383	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
4	3	anim1i 627	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋))
5		sseq2 3939	. . . . . 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 4008	. . 3 ⊢ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ (Clsd‘𝐽)
10		intcld 23388	. . 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 2864	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416   ⊆ wss 3900  ∅c0 4281  ∪ cuni 4870  ∩ cint 4910  ‘cfv 6558  Topctop 23241  Clsdccld 23364  clsccl 23366

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-top 23242  df-cld 23367  df-cls 23369

This theorem is used by:  clsf  23396  clsss3  23407  iscld3  23412  clsidm  23415  restcls  23529  cncls2i  23618  nrmsep  23705  lpcls  23712  regsep2  23724  hauscmplem  23754  hausllycmp  23843  txcls  23953  ptclsg  23964  regr1lem  24088  kqreglem1  24090  kqreglem2  24091  kqnrmlem1  24092  kqnrmlem2  24093  fclscmpi  24378  tgptsmscld  24500  cnllycmp  25307  clsocv  25601  cmpcmet  25670  cncmet  25673  limcnlp  26228  clsun  37294  cldregopn  37297  heibor1lem  38935  iscnrm3rlem2  50260  iscnrm3rlem5  50263