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Mirrors > Home > ILE Home > Th. List > restco | Unicode version |
Description: Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
restco |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2644 |
. . . . 5
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2 | 1 | inex1 4002 |
. . . 4
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3 | ineq1 3217 |
. . . . 5
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4 | inass 3233 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | syl6eq 2148 |
. . . 4
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6 | 2, 5 | abrexco 5592 |
. . 3
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7 | eqid 2100 |
. . . . . 6
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8 | 7 | rnmpt 4725 |
. . . . 5
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9 | mpteq1 3952 |
. . . . 5
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10 | 8, 9 | ax-mp 7 |
. . . 4
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11 | 10 | rnmpt 4725 |
. . 3
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12 | eqid 2100 |
. . . 4
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13 | 12 | rnmpt 4725 |
. . 3
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14 | 6, 11, 13 | 3eqtr4i 2130 |
. 2
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15 | restval 11908 |
. . . . 5
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16 | 15 | 3adant3 969 |
. . . 4
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17 | 16 | oveq1d 5721 |
. . 3
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18 | restfn 11906 |
. . . . . 6
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19 | simp1 949 |
. . . . . . 7
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20 | 19 | elexd 2654 |
. . . . . 6
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21 | simp2 950 |
. . . . . . 7
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22 | 21 | elexd 2654 |
. . . . . 6
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23 | fnovex 5736 |
. . . . . 6
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24 | 18, 20, 22, 23 | mp3an2i 1288 |
. . . . 5
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25 | 16, 24 | eqeltrrd 2177 |
. . . 4
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26 | simp3 951 |
. . . 4
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27 | restval 11908 |
. . . 4
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28 | 25, 26, 27 | syl2anc 406 |
. . 3
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29 | 17, 28 | eqtrd 2132 |
. 2
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30 | inex1g 4004 |
. . . 4
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31 | 30 | 3ad2ant2 971 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | restval 11908 |
. . 3
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33 | 19, 31, 32 | syl2anc 406 |
. 2
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34 | 14, 29, 33 | 3eqtr4a 2158 |
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-rest 11904 |
This theorem is referenced by: restabs 12126 restin 12127 |
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