Theorem restcldi 23132

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 1138	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (Clsd‘𝐽))
2		dfss 3922	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝐵 = (𝐵 ∩ 𝐴))
3	2	biimpi 216	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (𝐵 ∩ 𝐴))
4	3	3ad2ant3 1136	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 = (𝐵 ∩ 𝐴))
5		ineq1 4167	. . . 4 ⊢ (𝑣 = 𝐵 → (𝑣 ∩ 𝐴) = (𝐵 ∩ 𝐴))
6	5	rspceeqv 3601	. . 3 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 = (𝐵 ∩ 𝐴)) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴))
7	1, 4, 6	syl2anc 585	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴))
8		cldrcl 22985	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
9	8	3ad2ant2 1135	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐽 ∈ Top)
10		simp1 1137	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐴 ⊆ 𝑋)
11		restcldi.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
12	11	restcld 23131	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
13	9, 10, 12	syl2anc 585	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣 ∩ 𝐴)))
14	7, 13	mpbird 257	1 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (Clsd‘(𝐽 ↾t 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ∩ cin 3902   ⊆ wss 3903  ∪ cuni 4865  ‘cfv 6500  (class class class)co 7368   ↾_t crest 17352  Topctop 22852  Clsdccld 22975

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-en 8896  df-fin 8899  df-fi 9326  df-rest 17354  df-topgen 17375  df-top 22853  df-topon 22870  df-bases 22905  df-cld 22978

This theorem is referenced by:  txkgen  23611  qtoprest  23676  cnmpopc  24893  cnheiborlem  24924  abelth  26422  zarmxt1  34062  cvmliftlem10  35514  icccldii  49282