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Mirrors > Home > ILE Home > Th. List > cnrest | Unicode version |
Description: Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnrest.1 |
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Ref | Expression |
---|---|
cnrest |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnrest.1 |
. . . . 5
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2 | eqid 2100 |
. . . . 5
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3 | 1, 2 | cnf 12154 |
. . . 4
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4 | 3 | adantr 272 |
. . 3
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5 | simpr 109 |
. . 3
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6 | 4, 5 | fssresd 5235 |
. 2
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7 | cnvresima 4964 |
. . . 4
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8 | cntop1 12151 |
. . . . . . 7
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9 | 8 | adantr 272 |
. . . . . 6
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10 | 9 | adantr 272 |
. . . . 5
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11 | 1 | topopn 11957 |
. . . . . . . 8
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12 | ssexg 4007 |
. . . . . . . . 9
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13 | 12 | ancoms 266 |
. . . . . . . 8
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14 | 11, 13 | sylan 279 |
. . . . . . 7
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15 | 8, 14 | sylan 279 |
. . . . . 6
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16 | 15 | adantr 272 |
. . . . 5
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17 | cnima 12170 |
. . . . . 6
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18 | 17 | adantlr 464 |
. . . . 5
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19 | elrestr 11910 |
. . . . 5
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20 | 10, 16, 18, 19 | syl3anc 1184 |
. . . 4
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21 | 7, 20 | syl5eqel 2186 |
. . 3
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22 | 21 | ralrimiva 2464 |
. 2
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23 | 1 | toptopon 11967 |
. . . . 5
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24 | 8, 23 | sylib 121 |
. . . 4
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25 | resttopon 12122 |
. . . 4
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26 | 24, 25 | sylan 279 |
. . 3
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27 | cntop2 12152 |
. . . . 5
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28 | 27 | adantr 272 |
. . . 4
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29 | 2 | toptopon 11967 |
. . . 4
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30 | 28, 29 | sylib 121 |
. . 3
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31 | iscn 12147 |
. . 3
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32 | 26, 30, 31 | syl2anc 406 |
. 2
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33 | 6, 22, 32 | mpbir2and 896 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-map 6474 df-rest 11904 df-topgen 11923 df-top 11947 df-topon 11960 df-bases 11992 df-cn 12139 |
This theorem is referenced by: cnmpt1res 12246 cnmpt2res 12247 |
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