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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 20920 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | 1 | adantr 472 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ Top) |
3 | id 22 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋) | |
4 | toponmax 20932 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
5 | ssexg 4956 | . . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V) | |
6 | 3, 4, 5 | syl2anr 496 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
7 | resttop 21166 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top) | |
8 | 2, 6, 7 | syl2anc 696 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top) |
9 | simpr 479 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
10 | sseqin2 3960 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴) | |
11 | 9, 10 | sylib 208 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴) |
12 | simpl 474 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | |
13 | 4 | adantr 472 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽) |
14 | elrestr 16291 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
15 | 12, 6, 13, 14 | syl3anc 1477 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
16 | 11, 15 | eqeltrrd 2840 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
17 | elssuni 4619 | . . . 4 ⊢ (𝐴 ∈ (𝐽 ↾t 𝐴) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴)) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ (𝐽 ↾t 𝐴)) |
19 | restval 16289 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) | |
20 | 6, 19 | syldan 488 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) |
21 | inss2 3977 | . . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴 | |
22 | vex 3343 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
23 | 22 | inex1 4951 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V |
24 | 23 | elpw 4308 | . . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴) |
25 | 21, 24 | mpbir 221 | . . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴) |
27 | eqid 2760 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) | |
28 | 26, 27 | fmptd 6548 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴) |
29 | frn 6214 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴 → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴) | |
30 | 28, 29 | syl 17 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴) |
31 | 20, 30 | eqsstrd 3780 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴) |
32 | sspwuni 4763 | . . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) | |
33 | 31, 32 | sylib 208 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) |
34 | 18, 33 | eqssd 3761 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
35 | istopon 20919 | . 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴))) | |
36 | 8, 34, 35 | sylanbrc 701 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∩ cin 3714 ⊆ wss 3715 𝒫 cpw 4302 ∪ cuni 4588 ↦ cmpt 4881 ran crn 5267 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↾t crest 16283 Topctop 20900 TopOnctopon 20917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-oadd 7733 df-er 7911 df-en 8122 df-fin 8125 df-fi 8482 df-rest 16285 df-topgen 16306 df-top 20901 df-topon 20918 df-bases 20952 |
This theorem is referenced by: restuni 21168 stoig 21169 restsn2 21177 restlp 21189 restperf 21190 perfopn 21191 cnrest 21291 cnrest2 21292 cnrest2r 21293 cnpresti 21294 cnprest 21295 cnprest2 21296 restcnrm 21368 connsuba 21425 kgentopon 21543 1stckgenlem 21558 kgen2ss 21560 kgencn 21561 xkoinjcn 21692 qtoprest 21722 flimrest 21988 fclsrest 22029 flfcntr 22048 symgtgp 22106 dvrcn 22188 sszcld 22821 divcn 22872 cncfmptc 22915 cncfmptid 22916 cncfmpt2f 22918 cdivcncf 22921 cnmpt2pc 22928 icchmeo 22941 htpycc 22980 pcocn 23017 pcohtpylem 23019 pcopt 23022 pcopt2 23023 pcoass 23024 pcorevlem 23026 relcmpcmet 23315 limcvallem 23834 ellimc2 23840 limcres 23849 cnplimc 23850 cnlimc 23851 limccnp 23854 limccnp2 23855 dvbss 23864 perfdvf 23866 dvreslem 23872 dvres2lem 23873 dvcnp2 23882 dvcn 23883 dvaddbr 23900 dvmulbr 23901 dvcmulf 23907 dvmptres2 23924 dvmptcmul 23926 dvmptntr 23933 dvmptfsum 23937 dvcnvlem 23938 dvcnv 23939 lhop1lem 23975 lhop2 23977 lhop 23978 dvcnvrelem2 23980 dvcnvre 23981 ftc1lem3 24000 ftc1cn 24005 taylthlem1 24326 ulmdvlem3 24355 psercn 24379 abelth 24394 logcn 24592 cxpcn 24685 cxpcn2 24686 cxpcn3 24688 resqrtcn 24689 sqrtcn 24690 loglesqrt 24698 xrlimcnp 24894 efrlim 24895 ftalem3 25000 xrge0pluscn 30295 xrge0mulc1cn 30296 lmlimxrge0 30303 pnfneige0 30306 lmxrge0 30307 esumcvg 30457 cxpcncf1 30982 cvxpconn 31531 cvxsconn 31532 cvmsf1o 31561 cvmliftlem8 31581 cvmlift2lem9a 31592 cvmlift2lem11 31602 cvmlift3lem6 31613 ivthALT 32636 poimir 33755 broucube 33756 cnambfre 33771 ftc1cnnc 33797 areacirclem2 33814 areacirclem4 33816 fsumcncf 40594 ioccncflimc 40601 cncfuni 40602 icccncfext 40603 icocncflimc 40605 cncfiooicclem1 40609 cxpcncf2 40616 dvmptconst 40632 dvmptidg 40634 dvresntr 40635 itgsubsticclem 40694 dirkercncflem2 40824 dirkercncflem4 40826 fourierdlem32 40859 fourierdlem33 40860 fourierdlem62 40888 fourierdlem93 40919 fourierdlem101 40927 |
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