Theorem restcldi 21200

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 1132	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (Clsd‘𝐽))
2		dfss 3731	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
3	2	biimpi 206	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
4	3	3ad2ant3 1130	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 = (𝐵 ∩ 𝐴))
5		ineq1 3951	. . . . 5 ⊢ (𝑣 = 𝐵 → (𝑣 ∩ 𝐴) = (𝐵 ∩ 𝐴))
6	5	eqeq2d 2771	. . . 4 ⊢ (𝑣 = 𝐵 → (𝐵 = (𝑣 ∩ 𝐴) ↔ 𝐵 = (𝐵 ∩ 𝐴)))
7	6	rspcev 3450	. . 3 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 = (𝐵 ∩ 𝐴)) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴))
8	1, 4, 7	syl2anc 696	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴))
9		cldrcl 21053	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
10	9	3ad2ant2 1129	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top)
11		simp1 1131	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝑋)
12		restcldi.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
13	12	restcld 21199	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
14	10, 11, 13	syl2anc 696	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
15	8, 14	mpbird 247	1 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  ∃wrex 3052   ∩ cin 3715   ⊆ wss 3716  ∪ cuni 4589  ‘cfv 6050  (class class class)co 6815   ↾_t crest 16304  Topctop 20921  Clsdccld 21043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-oadd 7735  df-er 7914  df-en 8125  df-fin 8128  df-fi 8485  df-rest 16306  df-topgen 16327  df-top 20922  df-topon 20939  df-bases 20973  df-cld 21046

This theorem is referenced by:  txkgen  21678  qtoprest  21743  cnmpt2pc  22949  cnheiborlem  22975  abelth  24415  cvmliftlem10  31605