Theorem clsss 13251

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 3162	. . . . . 6 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥))
2	1	adantr 276	. . . . 5 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥))
3	2	ss2rabdv 3236	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
4		intss 3863	. . . 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 14	. . 3 ⊢ (𝑇 ⊆ 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
6	5	3ad2ant3 1020	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
7		simp1 997	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top)
8		sstr2 3162	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑋 → 𝑇 ⊆ 𝑋))
9	8	impcom 125	. . . 4 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
10	9	3adant1 1015	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
11		clscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
12	11	clsval 13244	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
13	7, 10, 12	syl2anc 411	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
14	11	clsval 13244	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
15	14	3adant3 1017	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
16	6, 13, 15	3sstr4d 3200	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {crab 2459   ⊆ wss 3129  ∪ cuni 3807  ∩ cint 3842  ‘cfv 5211  Topctop 13128  Clsdccld 13225  clsccl 13227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-top 13129  df-cld 13228  df-cls 13230

This theorem is referenced by:  clsss2  13262