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Mirrors > Home > MPE Home > Th. List > resttopon | Structured version Visualization version GIF version |
Description: A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
resttopon | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22934 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | id 22 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋) | |
3 | toponmax 22947 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
4 | ssexg 5328 | . . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | syl2anr 597 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
6 | resttop 23183 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top) | |
7 | 1, 5, 6 | syl2an2r 685 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top) |
8 | simpr 484 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
9 | sseqin2 4230 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴) | |
10 | 8, 9 | sylib 218 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴) |
11 | simpl 482 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | |
12 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽) |
13 | elrestr 17474 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
14 | 11, 5, 12, 13 | syl3anc 1370 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
15 | 10, 14 | eqeltrrd 2839 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
16 | elssuni 4941 | . . . 4 ⊢ (𝐴 ∈ (𝐽 ↾t 𝐴) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴)) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴)) |
18 | restval 17472 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) | |
19 | 5, 18 | syldan 591 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) |
20 | inss2 4245 | . . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴 | |
21 | vex 3481 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
22 | 21 | inex1 5322 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V |
23 | 22 | elpw 4608 | . . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴) |
24 | 20, 23 | mpbir 231 | . . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴) |
26 | 25 | fmpttd 7134 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴) |
27 | 26 | frnd 6744 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴) |
28 | 19, 27 | eqsstrd 4033 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴) |
29 | sspwuni 5104 | . . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) | |
30 | 28, 29 | sylib 218 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) |
31 | 17, 30 | eqssd 4012 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
32 | istopon 22933 | . 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴))) | |
33 | 7, 31, 32 | sylanbrc 583 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 𝒫 cpw 4604 ∪ cuni 4911 ↦ cmpt 5230 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ↾t crest 17466 Topctop 22914 TopOnctopon 22931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-en 8984 df-fin 8987 df-fi 9448 df-rest 17468 df-topgen 17489 df-top 22915 df-topon 22932 df-bases 22968 |
This theorem is referenced by: restuni 23185 stoig 23186 restsn2 23194 restlp 23206 restperf 23207 perfopn 23208 cnrest 23308 cnrest2 23309 cnrest2r 23310 cnpresti 23311 cnprest 23312 cnprest2 23313 restcnrm 23385 connsuba 23443 kgentopon 23561 1stckgenlem 23576 kgen2ss 23578 kgencn 23579 xkoinjcn 23710 qtoprest 23740 flimrest 24006 fclsrest 24047 flfcntr 24066 efmndtmd 24124 symgtgp 24129 dvrcn 24207 sszcld 24852 divcnOLD 24903 divcn 24905 cncfmptc 24951 cncfmptid 24952 cncfmpt2f 24954 cdivcncf 24960 cnmpopc 24968 icchmeo 24984 icchmeoOLD 24985 htpycc 25025 pcocn 25063 pcohtpylem 25065 pcopt 25068 pcopt2 25069 pcoass 25070 pcorevlem 25072 relcmpcmet 25365 mulcncf 25493 limcvallem 25920 ellimc2 25926 limcres 25935 cnplimc 25936 cnlimc 25937 limccnp 25940 limccnp2 25941 dvbss 25950 perfdvf 25952 dvreslem 25958 dvres2lem 25959 dvcnp2 25969 dvcnp2OLD 25970 dvcn 25971 dvaddbr 25988 dvmulbr 25989 dvmulbrOLD 25990 dvcmulf 25996 dvmptres2 26014 dvmptcmul 26016 dvmptntr 26023 dvmptfsum 26027 dvcnvlem 26028 dvcnv 26029 lhop1lem 26066 lhop2 26068 lhop 26069 dvcnvrelem2 26071 dvcnvre 26072 ftc1lem3 26093 ftc1cn 26098 taylthlem1 26429 ulmdvlem3 26459 psercn 26484 abelth 26499 logcn 26703 cxpcn 26801 cxpcnOLD 26802 cxpcn2 26803 cxpcn3 26805 resqrtcn 26806 sqrtcn 26807 loglesqrt 26818 xrlimcnp 27025 efrlim 27026 efrlimOLD 27027 ftalem3 27132 xrge0pluscn 33900 xrge0mulc1cn 33901 lmlimxrge0 33908 pnfneige0 33911 lmxrge0 33912 esumcvg 34066 cxpcncf1 34588 cvxpconn 35226 cvxsconn 35227 cvmsf1o 35256 cvmliftlem8 35276 cvmlift2lem9a 35287 cvmlift2lem11 35297 cvmlift3lem6 35308 ivthALT 36317 poimir 37639 broucube 37640 cnambfre 37654 ftc1cnnc 37678 areacirclem2 37695 areacirclem4 37697 fsumcncf 45833 ioccncflimc 45840 cncfuni 45841 icccncfext 45842 icocncflimc 45844 cncfiooicclem1 45848 cxpcncf2 45854 dvmptconst 45870 dvmptidg 45872 dvresntr 45873 itgsubsticclem 45930 dirkercncflem2 46059 dirkercncflem4 46061 fourierdlem32 46094 fourierdlem33 46095 fourierdlem62 46123 fourierdlem93 46154 fourierdlem101 46162 |
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