Theorem restcldi 23235

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

Hypothesis

Ref	Expression
restcldi.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
restcldi	⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)))

Proof of Theorem restcldi

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1151	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (Clsd‘𝐽))
2		dfss 3925	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
3	2	biimpi 218	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
4	3	3ad2ant3 1149	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 = (𝐵 ∩ 𝐴))
5		ineq1 4167	. . . 4 ⊢ (𝑣 = 𝐵 → (𝑣 ∩ 𝐴) = (𝐵 ∩ 𝐴))
6	5	rspceeqv 3606	. . 3 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 = (𝐵 ∩ 𝐴)) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴))
7	1, 4, 6	syl2anc 593	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴))
8		cldrcl 23088	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
9	8	3ad2ant2 1148	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top)
10		simp1 1150	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝑋)
11		restcldi.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
12	11	restcld 23234	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
13	9, 10, 12	syl2anc 593	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
14	7, 13	mpbird 259	1 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∃wrex 3088   ∩ cin 3905   ⊆ wss 3906  ∪ cuni 4867  ‘cfv 6523  (class class class)co 7398   ↾_t crest 17451  Topctop 22955  Clsdccld 23078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-en 8930  df-fin 8933  df-fi 9359  df-rest 17453  df-topgen 17474  df-top 22956  df-topon 22973  df-bases 23008  df-cld 23081

This theorem is referenced by:  txkgen  23714  qtoprest  23779  cnmpopc  24992  cnheiborlem  25018  abelth  26506  zarmxt1  34179  cvmliftlem10  35649  icccldii  49545