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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 22344 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | id 22 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋) | |
3 | toponmax 22357 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
4 | ssexg 5316 | . . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | syl2anr 597 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
6 | resttop 22593 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top) | |
7 | 1, 5, 6 | syl2an2r 683 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top) |
8 | simpr 485 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
9 | sseqin2 4211 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴) | |
10 | 8, 9 | sylib 217 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴) |
11 | simpl 483 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | |
12 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽) |
13 | elrestr 17356 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
14 | 11, 5, 12, 13 | syl3anc 1371 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
15 | 10, 14 | eqeltrrd 2833 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
16 | elssuni 4934 | . . . 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 17354 | . . . . . 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 4225 | . . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴 | |
21 | vex 3477 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
22 | 21 | inex1 5310 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V |
23 | 22 | elpw 4600 | . . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴) |
24 | 20, 23 | mpbir 230 | . . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴) |
26 | 25 | fmpttd 7099 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴) |
27 | 26 | frnd 6712 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴) |
28 | 19, 27 | eqsstrd 4016 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴) |
29 | sspwuni 5096 | . . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) | |
30 | 28, 29 | sylib 217 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) |
31 | 17, 30 | eqssd 3995 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
32 | istopon 22343 | . 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 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∩ cin 3943 ⊆ wss 3944 𝒫 cpw 4596 ∪ cuni 4901 ↦ cmpt 5224 ran crn 5670 ‘cfv 6532 (class class class)co 7393 ↾t crest 17348 Topctop 22324 TopOnctopon 22341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-en 8923 df-fin 8926 df-fi 9388 df-rest 17350 df-topgen 17371 df-top 22325 df-topon 22342 df-bases 22378 |
This theorem is referenced by: restuni 22595 stoig 22596 restsn2 22604 restlp 22616 restperf 22617 perfopn 22618 cnrest 22718 cnrest2 22719 cnrest2r 22720 cnpresti 22721 cnprest 22722 cnprest2 22723 restcnrm 22795 connsuba 22853 kgentopon 22971 1stckgenlem 22986 kgen2ss 22988 kgencn 22989 xkoinjcn 23120 qtoprest 23150 flimrest 23416 fclsrest 23457 flfcntr 23476 efmndtmd 23534 symgtgp 23539 dvrcn 23617 sszcld 24262 divcn 24313 cncfmptc 24357 cncfmptid 24358 cncfmpt2f 24360 cdivcncf 24366 cnmpopc 24373 icchmeo 24386 htpycc 24425 pcocn 24462 pcohtpylem 24464 pcopt 24467 pcopt2 24468 pcoass 24469 pcorevlem 24471 relcmpcmet 24764 limcvallem 25317 ellimc2 25323 limcres 25332 cnplimc 25333 cnlimc 25334 limccnp 25337 limccnp2 25338 dvbss 25347 perfdvf 25349 dvreslem 25355 dvres2lem 25356 dvcnp2 25366 dvcn 25367 dvaddbr 25384 dvmulbr 25385 dvcmulf 25391 dvmptres2 25408 dvmptcmul 25410 dvmptntr 25417 dvmptfsum 25421 dvcnvlem 25422 dvcnv 25423 lhop1lem 25459 lhop2 25461 lhop 25462 dvcnvrelem2 25464 dvcnvre 25465 ftc1lem3 25484 ftc1cn 25489 taylthlem1 25814 ulmdvlem3 25843 psercn 25867 abelth 25882 logcn 26084 cxpcn 26180 cxpcn2 26181 cxpcn3 26183 resqrtcn 26184 sqrtcn 26185 loglesqrt 26193 xrlimcnp 26400 efrlim 26401 ftalem3 26506 xrge0pluscn 32751 xrge0mulc1cn 32752 lmlimxrge0 32759 pnfneige0 32762 lmxrge0 32763 esumcvg 32915 cxpcncf1 33438 cvxpconn 34064 cvxsconn 34065 cvmsf1o 34094 cvmliftlem8 34114 cvmlift2lem9a 34125 cvmlift2lem11 34135 cvmlift3lem6 34146 ivthALT 35024 poimir 36325 broucube 36326 cnambfre 36340 ftc1cnnc 36364 areacirclem2 36381 areacirclem4 36383 fsumcncf 44367 ioccncflimc 44374 cncfuni 44375 icccncfext 44376 icocncflimc 44378 cncfiooicclem1 44382 cxpcncf2 44388 dvmptconst 44404 dvmptidg 44406 dvresntr 44407 itgsubsticclem 44464 dirkercncflem2 44593 dirkercncflem4 44595 fourierdlem32 44628 fourierdlem33 44629 fourierdlem62 44657 fourierdlem93 44688 fourierdlem101 44696 |
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