Theorem clscld 23087

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 23077	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
3	1	topcld 23075	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
4	3	anim1i 624	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋))
5		sseq2 3962	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋))
6	5	elrab 3650	. . . . 5 ⊢ (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋))
7	4, 6	sylibr 236	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
8	7	ne0d 4294	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ≠ ∅)
9		ssrab2 4033	. . 3 ⊢ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ (Clsd‘𝐽)
10		intcld 23080	. . 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 595	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ (Clsd‘𝐽))
12	2, 11	eqeltrd 2861	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  {crab 3413   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4864  ∩ cint 4904  ‘cfv 6517  Topctop 22933  Clsdccld 23056  clsccl 23058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-top 22934  df-cld 23059  df-cls 23061

This theorem is referenced by:  clsf  23088  clsss3  23099  iscld3  23104  clsidm  23107  restcls  23221  cncls2i  23310  nrmsep  23397  lpcls  23404  regsep2  23416  hauscmplem  23446  hausllycmp  23534  txcls  23644  ptclsg  23655  regr1lem  23779  kqreglem1  23781  kqreglem2  23782  kqnrmlem1  23783  kqnrmlem2  23784  fclscmpi  24069  tgptsmscld  24191  cnllycmp  24998  clsocv  25292  cmpcmet  25361  cncmet  25364  limcnlp  25920  clsun  36652  cldregopn  36655  heibor1lem  38272  iscnrm3rlem2  49526  iscnrm3rlem5  49529