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Mirrors > Home > MPE Home > Th. List > clsss | Structured version Visualization version GIF version |
Description: Subset relationship for closure. (Contributed by NM, 10-Feb-2007.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
clsss | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3610 | . . . . . 6 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥)) | |
2 | 1 | adantr 481 | . . . . 5 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ⊆ 𝑥 → 𝑇 ⊆ 𝑥)) |
3 | 2 | ss2rabdv 3683 | . . . 4 ⊢ (𝑇 ⊆ 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥}) |
4 | intss 4498 | . . . 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 17 | . . 3 ⊢ (𝑇 ⊆ 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
6 | 5 | 3ad2ant3 1084 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
7 | simp1 1061 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top) | |
8 | sstr2 3610 | . . . . 5 ⊢ (𝑇 ⊆ 𝑆 → (𝑆 ⊆ 𝑋 → 𝑇 ⊆ 𝑋)) | |
9 | 8 | impcom 446 | . . . 4 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋) |
10 | 9 | 3adant1 1079 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋) |
11 | clscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
12 | 11 | clsval 20841 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥}) |
13 | 7, 10, 12 | syl2anc 693 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑇 ⊆ 𝑥}) |
14 | 11 | clsval 20841 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
15 | 14 | 3adant3 1081 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
16 | 6, 13, 15 | 3sstr4d 3648 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘𝑇) ⊆ ((cls‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 ⊆ wss 3574 ∪ cuni 4436 ∩ cint 4475 ‘cfv 5888 Topctop 20698 Clsdccld 20820 clsccl 20822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-top 20699 df-cld 20823 df-cls 20825 |
This theorem is referenced by: ntrss 20859 clsss2 20876 lpsscls 20945 lpss3 20948 cnclsi 21076 cncls 21078 lpcls 21168 cnextcn 21871 clssubg 21912 clsnsg 21913 utopreg 22056 hauseqcn 29941 kur14lem6 31193 clsint2 32324 opnregcld 32325 |
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