Theorem clsss 12912

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 3154	. . . . . 6 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥))
2	1	adantr 274	. . . . 5 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥))
3	2	ss2rabdv 3228	. . . 4 ⊢ (𝑇 ⊆ 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
4		intss 3852	. . . 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 1015	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
7		simp1 992	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top)
8		sstr2 3154	. . . . 5 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑋 → 𝑇 ⊆ 𝑋))
9	8	impcom 124	. . . 4 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
10	9	3adant1 1010	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋)
11		clscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
12	11	clsval 12905	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
13	7, 10, 12	syl2anc 409	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥})
14	11	clsval 12905	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
15	14	3adant3 1012	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
16	6, 13, 15	3sstr4d 3192	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {crab 2452   ⊆ wss 3121  ∪ cuni 3796  ∩ cint 3831  ‘cfv 5198  Topctop 12789  Clsdccld 12886  clsccl 12888

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-cld 12889  df-cls 12891

This theorem is referenced by:  clsss2  12923