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Theorem clsss 12326
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 3109 . . . . . 6 (𝑇𝑆 → (𝑆𝑥𝑇𝑥))
21adantr 274 . . . . 5 ((𝑇𝑆𝑥 ∈ (Clsd‘𝐽)) → (𝑆𝑥𝑇𝑥))
32ss2rabdv 3183 . . . 4 (𝑇𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
4 intss 3800 . . . 4 ({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
53, 4syl 14 . . 3 (𝑇𝑆 {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
653ad2ant3 1005 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
7 simp1 982 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝐽 ∈ Top)
8 sstr2 3109 . . . . 5 (𝑇𝑆 → (𝑆𝑋𝑇𝑋))
98impcom 124 . . . 4 ((𝑆𝑋𝑇𝑆) → 𝑇𝑋)
1093adant1 1000 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → 𝑇𝑋)
11 clscld.1 . . . 4 𝑋 = 𝐽
1211clsval 12319 . . 3 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
137, 10, 12syl2anc 409 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇𝑥})
1411clsval 12319 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
15143adant3 1002 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑆) = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆𝑥})
166, 13, 153sstr4d 3147 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 963   = wceq 1332  wcel 1481  {crab 2421  wss 3076   cuni 3744   cint 3779  cfv 5131  Topctop 12203  Clsdccld 12300  clsccl 12302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-top 12204  df-cld 12303  df-cls 12305
This theorem is referenced by:  clsss2  12337
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