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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 2740 |
. . . . 5
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2 | 1 | inex1 4134 |
. . . 4
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3 | ineq1 3329 |
. . . . 5
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4 | inass 3345 |
. . . . 5
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5 | 3, 4 | eqtrdi 2226 |
. . . 4
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6 | 2, 5 | abrexco 5754 |
. . 3
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7 | eqid 2177 |
. . . . . 6
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8 | 7 | rnmpt 4871 |
. . . . 5
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9 | mpteq1 4084 |
. . . . 5
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10 | 8, 9 | ax-mp 5 |
. . . 4
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11 | 10 | rnmpt 4871 |
. . 3
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12 | eqid 2177 |
. . . 4
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13 | 12 | rnmpt 4871 |
. . 3
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14 | 6, 11, 13 | 3eqtr4i 2208 |
. 2
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15 | restval 12642 |
. . . . 5
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16 | 15 | 3adant3 1017 |
. . . 4
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17 | 16 | oveq1d 5884 |
. . 3
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18 | restfn 12640 |
. . . . . 6
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19 | simp1 997 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | elexd 2750 |
. . . . . 6
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21 | simp2 998 |
. . . . . . 7
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22 | 21 | elexd 2750 |
. . . . . 6
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23 | fnovex 5902 |
. . . . . 6
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24 | 18, 20, 22, 23 | mp3an2i 1342 |
. . . . 5
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25 | 16, 24 | eqeltrrd 2255 |
. . . 4
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26 | simp3 999 |
. . . 4
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27 | restval 12642 |
. . . 4
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28 | 25, 26, 27 | syl2anc 411 |
. . 3
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29 | 17, 28 | eqtrd 2210 |
. 2
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30 | inex1g 4136 |
. . . 4
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31 | 30 | 3ad2ant2 1019 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | restval 12642 |
. . 3
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33 | 19, 31, 32 | syl2anc 411 |
. 2
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34 | 14, 29, 33 | 3eqtr4a 2236 |
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 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 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-rest 12638 |
This theorem is referenced by: restabs 13342 restin 13343 |
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