Theorem restcldr 23061

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 22913	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		eqid 2729	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldss 22916	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
4	2	restcld 23059	. . . 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 22930	. . . . . 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 2816	. . . . 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 3134	. . 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 1540   ∈ wcel 2109  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387   ↾_t crest 17383  Topctop 22780  Clsdccld 22903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-en 8919  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-cld 22906

This theorem is referenced by:  paste  23181  qtoprest  23604  zcld2  24704  sszcld  24706  logdmopn  26558  dvasin  37698  dvacos  37699  dvreasin  37700  dvreacos  37701