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Mirrors > Home > ILE Home > Th. List > stoig | Unicode version |
Description: The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
restuni.1 |
Ref | Expression |
---|---|
stoig | TopSet ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restuni.1 | . . . 4 | |
2 | 1 | toptopon 12112 | . . 3 TopOn |
3 | resttopon 12267 | . . 3 TopOn ↾t TopOn | |
4 | 2, 3 | sylanb 282 | . 2 ↾t TopOn |
5 | eqid 2117 | . . 3 TopSet ↾t TopSet ↾t | |
6 | 5 | eltpsg 12134 | . 2 ↾t TopOn TopSet ↾t |
7 | 4, 6 | syl 14 | 1 TopSet ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wss 3041 cpr 3498 cop 3500 cuni 3706 cfv 5093 (class class class)co 5742 cnx 11883 cbs 11886 TopSetcts 11954 ↾t crest 12047 ctop 12091 TopOnctopon 12104 ctps 12124 |
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 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-pre-ltirr 7700 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
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-nel 2381 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-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 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-pnf 7770 df-mnf 7771 df-ltxr 7773 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-ndx 11889 df-slot 11890 df-base 11892 df-tset 11967 df-rest 12049 df-topn 12050 df-topgen 12068 df-top 12092 df-topon 12105 df-topsp 12125 df-bases 12137 |
This theorem is referenced by: (None) |
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