Theorem restcldr 23197

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 23049	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		eqid 2734	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldss 23052	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
4	2	restcld 23195	. . . 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 23066	. . . . . 6 ⊢ ((𝑣 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑣 ∩ 𝐴) ∈ (Clsd‘𝐽))
7	6	ancoms 458	. . . . 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 247	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝐵 = (𝑣 ∩ 𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
10	9	rexlimdva 3152	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
11	5, 10	sylbid 240	. 2 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) → 𝐵 ∈ (Clsd‘𝐽)))
12	11	imp 406	1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∃wrex 3067   ∩ cin 3961   ⊆ wss 3962  ∪ cuni 4911  ‘cfv 6562  (class class class)co 7430   ↾_t crest 17466  Topctop 22914  Clsdccld 23039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-en 8984  df-fin 8987  df-fi 9448  df-rest 17468  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-cld 23042

This theorem is referenced by:  paste  23317  qtoprest  23740  zcld2  24850  sszcld  24852  logdmopn  26705  dvasin  37690  dvacos  37691  dvreasin  37692  dvreacos  37693