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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 13991 |
. . . 4
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2 | 1 | adantr 276 |
. . 3
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3 | id 19 |
. . . 4
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4 | toponmax 14002 |
. . . 4
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5 | ssexg 4157 |
. . . 4
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6 | 3, 4, 5 | syl2anr 290 |
. . 3
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7 | resttop 14147 |
. . 3
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8 | 2, 6, 7 | syl2anc 411 |
. 2
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9 | simpr 110 |
. . . . . 6
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10 | sseqin2 3369 |
. . . . . 6
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11 | 9, 10 | sylib 122 |
. . . . 5
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12 | simpl 109 |
. . . . . 6
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13 | 4 | adantr 276 |
. . . . . 6
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14 | elrestr 12755 |
. . . . . 6
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15 | 12, 6, 13, 14 | syl3anc 1249 |
. . . . 5
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16 | 11, 15 | eqeltrrd 2267 |
. . . 4
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17 | elssuni 3852 |
. . . 4
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18 | 16, 17 | syl 14 |
. . 3
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19 | restval 12753 |
. . . . . 6
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20 | 6, 19 | syldan 282 |
. . . . 5
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21 | inss2 3371 |
. . . . . . . . 9
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22 | vex 2755 |
. . . . . . . . . . 11
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23 | 22 | inex1 4152 |
. . . . . . . . . 10
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24 | 23 | elpw 3596 |
. . . . . . . . 9
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25 | 21, 24 | mpbir 146 |
. . . . . . . 8
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26 | 25 | a1i 9 |
. . . . . . 7
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27 | 26 | fmpttd 5692 |
. . . . . 6
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28 | 27 | frnd 5394 |
. . . . 5
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29 | 20, 28 | eqsstrd 3206 |
. . . 4
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30 | sspwuni 3986 |
. . . 4
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31 | 29, 30 | sylib 122 |
. . 3
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32 | 18, 31 | eqssd 3187 |
. 2
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33 | istopon 13990 |
. 2
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34 | 8, 32, 33 | sylanbrc 417 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-rest 12749 df-topgen 12768 df-top 13975 df-topon 13988 df-bases 14020 |
This theorem is referenced by: restuni 14149 stoig 14150 cnrest 14212 cnrest2 14213 cnrest2r 14214 cnptopresti 14215 cnptoprest 14216 cnptoprest2 14217 divcnap 14532 cncfmpt2fcntop 14562 cnplimcim 14613 cnlimcim 14617 cnlimc 14618 limccnpcntop 14621 limccnp2lem 14622 limccnp2cntop 14623 dvfvalap 14627 dvbss 14631 dvfgg 14634 dvcnp2cntop 14640 dvcn 14641 dvaddxxbr 14642 dvmulxxbr 14643 |
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