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Mirrors > Home > ILE Home > Th. List > hmeores | Unicode version |
Description: The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
hmeores.1 |
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Ref | Expression |
---|---|
hmeores |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocn 13808 |
. . . . 5
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2 | 1 | adantr 276 |
. . . 4
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3 | hmeores.1 |
. . . . 5
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4 | 3 | cnrest 13738 |
. . . 4
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5 | 2, 4 | sylancom 420 |
. . 3
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6 | cntop2 13705 |
. . . . . 6
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7 | 2, 6 | syl 14 |
. . . . 5
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8 | eqid 2177 |
. . . . . 6
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9 | 8 | toptopon 13521 |
. . . . 5
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10 | 7, 9 | sylib 122 |
. . . 4
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11 | df-ima 4640 |
. . . . . 6
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12 | 11 | eqimss2i 3213 |
. . . . 5
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13 | 12 | a1i 9 |
. . . 4
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14 | imassrn 4982 |
. . . . 5
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15 | 3, 8 | cnf 13707 |
. . . . . . 7
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16 | 2, 15 | syl 14 |
. . . . . 6
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17 | 16 | frnd 5376 |
. . . . 5
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18 | 14, 17 | sstrid 3167 |
. . . 4
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19 | cnrest2 13739 |
. . . 4
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20 | 10, 13, 18, 19 | syl3anc 1238 |
. . 3
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21 | 5, 20 | mpbid 147 |
. 2
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22 | hmeocnvcn 13809 |
. . . . . 6
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23 | 22 | adantr 276 |
. . . . 5
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24 | 8, 3 | cnf 13707 |
. . . . 5
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25 | ffun 5369 |
. . . . 5
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26 | funcnvres 5290 |
. . . . 5
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27 | 23, 24, 25, 26 | 4syl 18 |
. . . 4
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28 | 8 | cnrest 13738 |
. . . . 5
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29 | 23, 18, 28 | syl2anc 411 |
. . . 4
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30 | 27, 29 | eqeltrd 2254 |
. . 3
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31 | cntop1 13704 |
. . . . . 6
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32 | 2, 31 | syl 14 |
. . . . 5
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33 | 3 | toptopon 13521 |
. . . . 5
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34 | 32, 33 | sylib 122 |
. . . 4
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35 | dfdm4 4820 |
. . . . . 6
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36 | fssres 5392 |
. . . . . . . 8
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37 | 16, 36 | sylancom 420 |
. . . . . . 7
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38 | 37 | fdmd 5373 |
. . . . . 6
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39 | 35, 38 | eqtr3id 2224 |
. . . . 5
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40 | eqimss 3210 |
. . . . 5
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41 | 39, 40 | syl 14 |
. . . 4
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42 | simpr 110 |
. . . 4
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43 | cnrest2 13739 |
. . . 4
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44 | 34, 41, 42, 43 | syl3anc 1238 |
. . 3
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45 | 30, 44 | mpbid 147 |
. 2
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46 | ishmeo 13807 |
. 2
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47 | 21, 45, 46 | sylanbrc 417 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-map 6650 df-rest 12690 df-topgen 12709 df-top 13501 df-topon 13514 df-bases 13546 df-cn 13691 df-hmeo 13804 |
This theorem is referenced by: (None) |
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