Theorem restcldr 22325

Description: A set which is closed in the subspace topology induced by a closed set is closed in the original topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
restcldr	⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))

Proof of Theorem restcldr

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldrcl 22177	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		eqid 2738	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldss 22180	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
4	2	restcld 22323	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
5	1, 3, 4	syl2anc 584	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
6		incld 22194	. . . . . 6 ⊢ ((𝑣 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑣 ∩ 𝐴) ∈ (Clsd‘𝐽))
7	6	ancoms 459	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝑣 ∩ 𝐴) ∈ (Clsd‘𝐽))
8		eleq1 2826	. . . . 5 ⊢ (𝐵 = (𝑣 ∩ 𝐴) → (𝐵 ∈ (Clsd‘𝐽) ↔ (𝑣 ∩ 𝐴) ∈ (Clsd‘𝐽)))
9	7, 8	syl5ibrcom 246	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝐵 = (𝑣 ∩ 𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
10	9	rexlimdva 3213	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
11	5, 10	sylbid 239	. 2 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) → 𝐵 ∈ (Clsd‘𝐽)))
12	11	imp 407	1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  ∪ cuni 4839  ‘cfv 6433  (class class class)co 7275   ↾_t crest 17131  Topctop 22042  Clsdccld 22167

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-3or 1087  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-pss 3906  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-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  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-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  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-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170

This theorem is referenced by:  paste  22445  qtoprest  22868  zcld2  23978  sszcld  23980  logdmopn  25804  dvasin  35861  dvacos  35862  dvreasin  35863  dvreacos  35864