Theorem restcldr 23525

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 23377	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		eqid 2763	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldss 23380	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽)
4	2	restcld 23523	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
5	1, 3, 4	syl2anc 596	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
6		incld 23394	. . . . . 6 ⊢ ((𝑣 ∈ (Clsd‘𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝑣 ∩ 𝐴) ∈ (Clsd‘𝐽))
7	6	ancoms 464	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝑣 ∩ 𝐴) ∈ (Clsd‘𝐽))
8		eleq1 2852	. . . . 5 ⊢ (𝐵 = (𝑣 ∩ 𝐴) → (𝐵 ∈ (Clsd‘𝐽) ↔ (𝑣 ∩ 𝐴) ∈ (Clsd‘𝐽)))
9	7, 8	syl5ibrcom 250	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝑣 ∈ (Clsd‘𝐽)) → (𝐵 = (𝑣 ∩ 𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
10	9	rexlimdva 3167	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴) → 𝐵 ∈ (Clsd‘𝐽)))
11	5, 10	sylbid 243	. 2 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) → 𝐵 ∈ (Clsd‘𝐽)))
12	11	imp 412	1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴))) → 𝐵 ∈ (Clsd‘𝐽))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ∪ cuni 4870  ‘cfv 6561  (class class class)co 7446   ↾_t crest 17623  Topctop 23244  Clsdccld 23367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-en 9005  df-fin 9008  df-fi 9433  df-rest 17625  df-topgen 17646  df-top 23245  df-topon 23262  df-bases 23297  df-cld 23370

This theorem is used by:  paste  23645  qtoprest  24069  zcld2  25168  sszcld  25170  logdmopn  27010  dvasin  38827  dvacos  38828  dvreasin  38829  dvreacos  38830