Theorem clsss 23176

Description: Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsss	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem clsss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3952	. . . . . 6 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥))
2	1	adantr 485	. . . . 5 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥))
3	2	ss2rabdv 4037	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
4		intss 4935	. . . 4 ⊢ ({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
5	3, 4	syl 18	. . 3 ⊢ (𝑇 ⊆ 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
6	5	3ad2ant3 1151	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
7		simp1 1152	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top)
8		sstr2 3952	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑋 → 𝑇 ⊆ 𝑋))
9	8	impcom 412	. . . 4 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
10	9	3adant1 1146	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
11		clscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
12	11	clsval 23159	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
13	7, 10, 12	syl2anc 595	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
14	11	clsval 23159	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
15	14	3adant3 1148	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
16	6, 13, 15	3sstr4d 4000	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {crab 3423   ⊆ wss 3913  ∪ cuni 4873  ∩ cint 4913  ‘cfv 6533  Topctop 23015  Clsdccld 23138  clsccl 23140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-top 23016  df-cld 23141  df-cls 23143

This theorem is referenced by:  ntrss  23177  clsss2  23194  lpsscls  23263  lpss3  23266  cnclsi  23394  cncls  23396  lpcls  23486  cnextcn  24189  clssubg  24231  clsnsg  24232  utopreg  24374  hauseqcn  34229  kur14lem6  35598  clsint2  36725  opnregcld  36726