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Theorem clsss 22957
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 3987 . . . . . 6 (𝑇𝑆 → (𝑆𝑥𝑇𝑥))
21adantr 480 . . . . 5 ((𝑇𝑆𝑥 ∈ (Clsd‘𝐽)) → (𝑆𝑥𝑇𝑥))
32ss2rabdv 4071 . . . 4 (𝑇𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
4 intss 4972 . . . 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 1133 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
7 simp1 1134 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝐽 ∈ Top)
8 sstr2 3987 . . . . 5 (𝑇𝑆 → (𝑆𝑋𝑇𝑋))
98impcom 407 . . . 4 ((𝑆𝑋𝑇𝑆) → 𝑇𝑋)
1093adant1 1128 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑋)
11 clscld.1 . . . 4 𝑋 = 𝐽
1211clsval 22940 . . 3 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
137, 10, 12syl2anc 583 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
1411clsval 22940 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
15143adant3 1130 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
166, 13, 153sstr4d 4027 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  {crab 3429  wss 3947   cuni 4908   cint 4949  cfv 6548  Topctop 22794  Clsdccld 22919  clsccl 22921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-top 22795  df-cld 22922  df-cls 22924
This theorem is referenced by:  ntrss  22958  clsss2  22975  lpsscls  23044  lpss3  23047  cnclsi  23175  cncls  23177  lpcls  23267  cnextcn  23970  clssubg  24012  clsnsg  24013  utopreg  24156  hauseqcn  33499  kur14lem6  34821  clsint2  35813  opnregcld  35814
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