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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 13775 |
. . . . 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 13705 |
. . . 4
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5 | 2, 4 | sylancom 420 |
. . 3
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6 | cntop2 13672 |
. . . . . 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 13488 |
. . . . 5
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10 | 7, 9 | sylib 122 |
. . . 4
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11 | df-ima 4639 |
. . . . . 6
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12 | 11 | eqimss2i 3212 |
. . . . 5
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13 | 12 | a1i 9 |
. . . 4
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14 | imassrn 4981 |
. . . . 5
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15 | 3, 8 | cnf 13674 |
. . . . . . 7
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16 | 2, 15 | syl 14 |
. . . . . 6
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17 | 16 | frnd 5375 |
. . . . 5
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18 | 14, 17 | sstrid 3166 |
. . . 4
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19 | cnrest2 13706 |
. . . 4
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20 | 10, 13, 18, 19 | syl3anc 1238 |
. . 3
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21 | 5, 20 | mpbid 147 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | hmeocnvcn 13776 |
. . . . . 6
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23 | 22 | adantr 276 |
. . . . 5
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24 | 8, 3 | cnf 13674 |
. . . . 5
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25 | ffun 5368 |
. . . . 5
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26 | funcnvres 5289 |
. . . . 5
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27 | 23, 24, 25, 26 | 4syl 18 |
. . . 4
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28 | 8 | cnrest 13705 |
. . . . 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 13671 |
. . . . . 6
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32 | 2, 31 | syl 14 |
. . . . 5
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33 | 3 | toptopon 13488 |
. . . . 5
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34 | 32, 33 | sylib 122 |
. . . 4
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35 | dfdm4 4819 |
. . . . . 6
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36 | fssres 5391 |
. . . . . . . 8
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37 | 16, 36 | sylancom 420 |
. . . . . . 7
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38 | 37 | fdmd 5372 |
. . . . . 6
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39 | 35, 38 | eqtr3id 2224 |
. . . . 5
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40 | eqimss 3209 |
. . . . 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 13706 |
. . . 4
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44 | 34, 41, 42, 43 | syl3anc 1238 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 30, 44 | mpbid 147 |
. 2
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46 | ishmeo 13774 |
. 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 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 |
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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-map 6649 df-rest 12689 df-topgen 12708 df-top 13468 df-topon 13481 df-bases 13513 df-cn 13658 df-hmeo 13771 |
This theorem is referenced by: (None) |
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