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Mirrors > Home > ILE Home > Th. List > resttopon | Unicode version |
Description: A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
resttopon | TopOn ↾t TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 12092 | . . . 4 TopOn | |
2 | 1 | adantr 274 | . . 3 TopOn |
3 | id 19 | . . . 4 | |
4 | toponmax 12103 | . . . 4 TopOn | |
5 | ssexg 4037 | . . . 4 | |
6 | 3, 4, 5 | syl2anr 288 | . . 3 TopOn |
7 | resttop 12250 | . . 3 ↾t | |
8 | 2, 6, 7 | syl2anc 408 | . 2 TopOn ↾t |
9 | simpr 109 | . . . . . 6 TopOn | |
10 | sseqin2 3265 | . . . . . 6 | |
11 | 9, 10 | sylib 121 | . . . . 5 TopOn |
12 | simpl 108 | . . . . . 6 TopOn TopOn | |
13 | 4 | adantr 274 | . . . . . 6 TopOn |
14 | elrestr 12039 | . . . . . 6 TopOn ↾t | |
15 | 12, 6, 13, 14 | syl3anc 1201 | . . . . 5 TopOn ↾t |
16 | 11, 15 | eqeltrrd 2195 | . . . 4 TopOn ↾t |
17 | elssuni 3734 | . . . 4 ↾t ↾t | |
18 | 16, 17 | syl 14 | . . 3 TopOn ↾t |
19 | restval 12037 | . . . . . 6 TopOn ↾t | |
20 | 6, 19 | syldan 280 | . . . . 5 TopOn ↾t |
21 | inss2 3267 | . . . . . . . . 9 | |
22 | vex 2663 | . . . . . . . . . . 11 | |
23 | 22 | inex1 4032 | . . . . . . . . . 10 |
24 | 23 | elpw 3486 | . . . . . . . . 9 |
25 | 21, 24 | mpbir 145 | . . . . . . . 8 |
26 | 25 | a1i 9 | . . . . . . 7 TopOn |
27 | 26 | fmpttd 5543 | . . . . . 6 TopOn |
28 | 27 | frnd 5252 | . . . . 5 TopOn |
29 | 20, 28 | eqsstrd 3103 | . . . 4 TopOn ↾t |
30 | sspwuni 3867 | . . . 4 ↾t ↾t | |
31 | 29, 30 | sylib 121 | . . 3 TopOn ↾t |
32 | 18, 31 | eqssd 3084 | . 2 TopOn ↾t |
33 | istopon 12091 | . 2 ↾t TopOn ↾t ↾t | |
34 | 8, 32, 33 | sylanbrc 413 | 1 TopOn ↾t TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cvv 2660 cin 3040 wss 3041 cpw 3480 cuni 3706 cmpt 3959 crn 4510 cfv 5093 (class class class)co 5742 ↾t crest 12031 ctop 12075 TopOnctopon 12088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-rest 12033 df-topgen 12052 df-top 12076 df-topon 12089 df-bases 12121 |
This theorem is referenced by: restuni 12252 stoig 12253 cnrest 12315 cnrest2 12316 cnrest2r 12317 cnptopresti 12318 cnptoprest 12319 cnptoprest2 12320 divcnap 12635 cncfmpt2fcntop 12665 cnplimcim 12716 cnlimcim 12720 cnlimc 12721 limccnpcntop 12724 limccnp2lem 12725 limccnp2cntop 12726 dvfvalap 12730 dvbss 12734 dvfgg 12737 dvcnp2cntop 12743 dvcn 12744 dvaddxxbr 12745 dvmulxxbr 12746 |
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