Theorem restcldr 23149

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 23001	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldss 23004	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
4	2	restcld 23147	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
5	1, 3, 4	syl2anc 585	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
6		incld 23018	. . . . . 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 2825	. . . . 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 3139	. . 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 1542   ∈ wcel 2114  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  ∪ cuni 4851  ‘cfv 6492  (class class class)co 7360   ↾_t crest 17374  Topctop 22868  Clsdccld 22991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-en 8887  df-fin 8890  df-fi 9317  df-rest 17376  df-topgen 17397  df-top 22869  df-topon 22886  df-bases 22921  df-cld 22994

This theorem is referenced by:  paste  23269  qtoprest  23692  zcld2  24791  sszcld  24793  logdmopn  26626  dvasin  38039  dvacos  38040  dvreasin  38041  dvreacos  38042