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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 12220 |
. . . 4
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2 | 1 | adantr 274 |
. . 3
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3 | id 19 |
. . . 4
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4 | toponmax 12231 |
. . . 4
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5 | ssexg 4075 |
. . . 4
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6 | 3, 4, 5 | syl2anr 288 |
. . 3
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7 | resttop 12378 |
. . 3
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8 | 2, 6, 7 | syl2anc 409 |
. 2
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9 | simpr 109 |
. . . . . 6
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10 | sseqin2 3300 |
. . . . . 6
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11 | 9, 10 | sylib 121 |
. . . . 5
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12 | simpl 108 |
. . . . . 6
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13 | 4 | adantr 274 |
. . . . . 6
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14 | elrestr 12167 |
. . . . . 6
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15 | 12, 6, 13, 14 | syl3anc 1217 |
. . . . 5
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16 | 11, 15 | eqeltrrd 2218 |
. . . 4
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17 | elssuni 3772 |
. . . 4
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18 | 16, 17 | syl 14 |
. . 3
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19 | restval 12165 |
. . . . . 6
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20 | 6, 19 | syldan 280 |
. . . . 5
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21 | inss2 3302 |
. . . . . . . . 9
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22 | vex 2692 |
. . . . . . . . . . 11
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23 | 22 | inex1 4070 |
. . . . . . . . . 10
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24 | 23 | elpw 3521 |
. . . . . . . . 9
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25 | 21, 24 | mpbir 145 |
. . . . . . . 8
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26 | 25 | a1i 9 |
. . . . . . 7
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27 | 26 | fmpttd 5583 |
. . . . . 6
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28 | 27 | frnd 5290 |
. . . . 5
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29 | 20, 28 | eqsstrd 3138 |
. . . 4
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30 | sspwuni 3905 |
. . . 4
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31 | 29, 30 | sylib 121 |
. . 3
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32 | 18, 31 | eqssd 3119 |
. 2
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33 | istopon 12219 |
. 2
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34 | 8, 32, 33 | sylanbrc 414 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-rest 12161 df-topgen 12180 df-top 12204 df-topon 12217 df-bases 12249 |
This theorem is referenced by: restuni 12380 stoig 12381 cnrest 12443 cnrest2 12444 cnrest2r 12445 cnptopresti 12446 cnptoprest 12447 cnptoprest2 12448 divcnap 12763 cncfmpt2fcntop 12793 cnplimcim 12844 cnlimcim 12848 cnlimc 12849 limccnpcntop 12852 limccnp2lem 12853 limccnp2cntop 12854 dvfvalap 12858 dvbss 12862 dvfgg 12865 dvcnp2cntop 12871 dvcn 12872 dvaddxxbr 12873 dvmulxxbr 12874 |
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