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Theorem clsss 23062
Description: Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clsss ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem clsss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sstr2 3990 . . . . . 6 (𝑇𝑆 → (𝑆𝑥𝑇𝑥))
21adantr 480 . . . . 5 ((𝑇𝑆𝑥 ∈ (Clsd‘𝐽)) → (𝑆𝑥𝑇𝑥))
32ss2rabdv 4076 . . . 4 (𝑇𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
4 intss 4969 . . . 4 ({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
53, 4syl 17 . . 3 (𝑇𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
653ad2ant3 1136 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
7 simp1 1137 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝐽 ∈ Top)
8 sstr2 3990 . . . . 5 (𝑇𝑆 → (𝑆𝑋𝑇𝑋))
98impcom 407 . . . 4 ((𝑆𝑋𝑇𝑆) → 𝑇𝑋)
1093adant1 1131 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑋)
11 clscld.1 . . . 4 𝑋 = 𝐽
1211clsval 23045 . . 3 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
137, 10, 12syl2anc 584 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
1411clsval 23045 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
15143adant3 1133 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
166, 13, 153sstr4d 4039 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  wss 3951   cuni 4907   cint 4946  cfv 6561  Topctop 22899  Clsdccld 23024  clsccl 23026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-cls 23029
This theorem is referenced by:  ntrss  23063  clsss2  23080  lpsscls  23149  lpss3  23152  cnclsi  23280  cncls  23282  lpcls  23372  cnextcn  24075  clssubg  24117  clsnsg  24118  utopreg  24261  hauseqcn  33897  kur14lem6  35216  clsint2  36330  opnregcld  36331
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