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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 21521 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | id 22 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋) | |
3 | toponmax 21534 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
4 | ssexg 5227 | . . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | syl2anr 598 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
6 | resttop 21768 | . . 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 487 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
9 | sseqin2 4192 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴) | |
10 | 8, 9 | sylib 220 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴) |
11 | simpl 485 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | |
12 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽) |
13 | elrestr 16702 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
14 | 11, 5, 12, 13 | syl3anc 1367 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
15 | 10, 14 | eqeltrrd 2914 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
16 | elssuni 4868 | . . . 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 16700 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) | |
19 | 5, 18 | syldan 593 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) |
20 | inss2 4206 | . . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴 | |
21 | vex 3497 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
22 | 21 | inex1 5221 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V |
23 | 22 | elpw 4543 | . . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴) |
24 | 20, 23 | mpbir 233 | . . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴) |
26 | 25 | fmpttd 6879 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴) |
27 | 26 | frnd 6521 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ⊆ 𝒫 𝐴) |
28 | 19, 27 | eqsstrd 4005 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴) |
29 | sspwuni 5022 | . . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) | |
30 | 28, 29 | sylib 220 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) |
31 | 17, 30 | eqssd 3984 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
32 | istopon 21520 | . 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴))) | |
33 | 7, 31, 32 | sylanbrc 585 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ↾t crest 16694 Topctop 21501 TopOnctopon 21518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 df-fi 8875 df-rest 16696 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 |
This theorem is referenced by: restuni 21770 stoig 21771 restsn2 21779 restlp 21791 restperf 21792 perfopn 21793 cnrest 21893 cnrest2 21894 cnrest2r 21895 cnpresti 21896 cnprest 21897 cnprest2 21898 restcnrm 21970 connsuba 22028 kgentopon 22146 1stckgenlem 22161 kgen2ss 22163 kgencn 22164 xkoinjcn 22295 qtoprest 22325 flimrest 22591 fclsrest 22632 flfcntr 22651 efmndtmd 22709 symgtgp 22714 dvrcn 22792 sszcld 23425 divcn 23476 cncfmptc 23519 cncfmptid 23520 cncfmpt2f 23522 cdivcncf 23525 cnmpopc 23532 icchmeo 23545 htpycc 23584 pcocn 23621 pcohtpylem 23623 pcopt 23626 pcopt2 23627 pcoass 23628 pcorevlem 23630 relcmpcmet 23921 limcvallem 24469 ellimc2 24475 limcres 24484 cnplimc 24485 cnlimc 24486 limccnp 24489 limccnp2 24490 dvbss 24499 perfdvf 24501 dvreslem 24507 dvres2lem 24508 dvcnp2 24517 dvcn 24518 dvaddbr 24535 dvmulbr 24536 dvcmulf 24542 dvmptres2 24559 dvmptcmul 24561 dvmptntr 24568 dvmptfsum 24572 dvcnvlem 24573 dvcnv 24574 lhop1lem 24610 lhop2 24612 lhop 24613 dvcnvrelem2 24615 dvcnvre 24616 ftc1lem3 24635 ftc1cn 24640 taylthlem1 24961 ulmdvlem3 24990 psercn 25014 abelth 25029 logcn 25230 cxpcn 25326 cxpcn2 25327 cxpcn3 25329 resqrtcn 25330 sqrtcn 25331 loglesqrt 25339 xrlimcnp 25546 efrlim 25547 ftalem3 25652 xrge0pluscn 31183 xrge0mulc1cn 31184 lmlimxrge0 31191 pnfneige0 31194 lmxrge0 31195 esumcvg 31345 cxpcncf1 31866 cvxpconn 32489 cvxsconn 32490 cvmsf1o 32519 cvmliftlem8 32539 cvmlift2lem9a 32550 cvmlift2lem11 32560 cvmlift3lem6 32571 ivthALT 33683 poimir 34940 broucube 34941 cnambfre 34955 ftc1cnnc 34981 areacirclem2 34998 areacirclem4 35000 fsumcncf 42181 ioccncflimc 42188 cncfuni 42189 icccncfext 42190 icocncflimc 42192 cncfiooicclem1 42196 cxpcncf2 42203 dvmptconst 42219 dvmptidg 42221 dvresntr 42222 itgsubsticclem 42280 dirkercncflem2 42409 dirkercncflem4 42411 fourierdlem32 42444 fourierdlem33 42445 fourierdlem62 42473 fourierdlem93 42504 fourierdlem101 42512 |
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