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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 20541 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ Top) |
3 | id 22 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ 𝑋) | |
4 | toponmax 20543 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
5 | ssexg 4732 | . . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐴 ∈ V) | |
6 | 3, 4, 5 | syl2anr 494 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
7 | resttop 20774 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ Top) | |
8 | 2, 6, 7 | syl2anc 691 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ Top) |
9 | simpr 476 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) | |
10 | sseqin2 3779 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴) | |
11 | 9, 10 | sylib 207 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴) |
12 | simpl 472 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | |
13 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ 𝐽) |
14 | elrestr 15912 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V ∧ 𝑋 ∈ 𝐽) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) | |
15 | 12, 6, 13, 14 | syl3anc 1318 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑋 ∩ 𝐴) ∈ (𝐽 ↾t 𝐴)) |
16 | 11, 15 | eqeltrrd 2689 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ (𝐽 ↾t 𝐴)) |
17 | elssuni 4403 | . . . 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 15910 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) | |
20 | 6, 19 | syldan 486 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴))) |
21 | inss2 3796 | . . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴 | |
22 | vex 3176 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
23 | 22 | inex1 4727 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ∈ V |
24 | 23 | elpw 4114 | . . . . . . . . 9 ⊢ ((𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝐴) ⊆ 𝐴) |
25 | 21, 24 | mpbir 220 | . . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴 |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ 𝒫 𝐴) |
27 | eqid 2610 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) | |
28 | 26, 27 | fmptd 6292 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)):𝐽⟶𝒫 𝐴) |
29 | frn 5966 | . . . . . 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 3602 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴) |
32 | sspwuni 4547 | . . . 4 ⊢ ((𝐽 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) | |
33 | 31, 32 | sylib 207 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ∪ (𝐽 ↾t 𝐴) ⊆ 𝐴) |
34 | 18, 33 | eqssd 3585 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴)) |
35 | istopon 20540 | . 2 ⊢ ((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ↔ ((𝐽 ↾t 𝐴) ∈ Top ∧ 𝐴 = ∪ (𝐽 ↾t 𝐴))) | |
36 | 8, 34, 35 | sylanbrc 695 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↾t crest 15904 Topctop 20517 TopOnctopon 20518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 df-fi 8200 df-rest 15906 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 |
This theorem is referenced by: restuni 20776 stoig 20777 restsn2 20785 restlp 20797 restperf 20798 perfopn 20799 cnrest 20899 cnrest2 20900 cnrest2r 20901 cnpresti 20902 cnprest 20903 cnprest2 20904 restcnrm 20976 consuba 21033 kgentopon 21151 1stckgenlem 21166 kgen2ss 21168 kgencn 21169 xkoinjcn 21300 qtoprest 21330 flimrest 21597 fclsrest 21638 flfcntr 21657 symgtgp 21715 dvrcn 21797 sszcld 22428 divcn 22479 cncfmptc 22522 cncfmptid 22523 cncfmpt2f 22525 cdivcncf 22528 cnmpt2pc 22535 icchmeo 22548 htpycc 22587 pcocn 22625 pcohtpylem 22627 pcopt 22630 pcopt2 22631 pcoass 22632 pcorevlem 22634 relcmpcmet 22923 limcvallem 23441 ellimc2 23447 limcres 23456 cnplimc 23457 cnlimc 23458 limccnp 23461 limccnp2 23462 dvbss 23471 perfdvf 23473 dvreslem 23479 dvres2lem 23480 dvcnp2 23489 dvcn 23490 dvaddbr 23507 dvmulbr 23508 dvcmulf 23514 dvmptres2 23531 dvmptcmul 23533 dvmptntr 23540 dvmptfsum 23542 dvcnvlem 23543 dvcnv 23544 lhop1lem 23580 lhop2 23582 lhop 23583 dvcnvrelem2 23585 dvcnvre 23586 ftc1lem3 23605 ftc1cn 23610 taylthlem1 23931 ulmdvlem3 23960 psercn 23984 abelth 23999 logcn 24193 cxpcn 24286 cxpcn2 24287 cxpcn3 24289 resqrtcn 24290 sqrtcn 24291 loglesqrt 24299 xrlimcnp 24495 efrlim 24496 ftalem3 24601 xrge0pluscn 29314 xrge0mulc1cn 29315 lmlimxrge0 29322 pnfneige0 29325 lmxrge0 29326 esumcvg 29475 cvxpcon 30478 cvxscon 30479 cvmsf1o 30508 cvmliftlem8 30528 cvmlift2lem9a 30539 cvmlift2lem11 30549 cvmlift3lem6 30560 ivthALT 31500 poimir 32612 broucube 32613 cnambfre 32628 ftc1cnnc 32654 areacirclem2 32671 areacirclem4 32673 fsumcncf 38763 ioccncflimc 38771 cncfuni 38772 icccncfext 38773 icocncflimc 38775 cncfiooicclem1 38779 cxpcncf2 38786 dvmptconst 38803 dvmptidg 38805 dvresntr 38806 itgsubsticclem 38867 dirkercncflem2 38997 dirkercncflem4 38999 fourierdlem32 39032 fourierdlem33 39033 fourierdlem62 39061 fourierdlem93 39092 fourierdlem101 39100 |
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