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Mirrors > Home > MPE Home > Th. List > restco | Structured version Visualization version 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 3060 |
. . . . 5
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2 | 1 | inex1 4558 |
. . . 4
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3 | ineq1 3639 |
. . . . 5
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4 | inass 3654 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | syl6eq 2512 |
. . . 4
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6 | 2, 5 | abrexco 6174 |
. . 3
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7 | eqid 2462 |
. . . . . 6
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8 | 7 | rnmpt 5099 |
. . . . 5
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9 | mpteq1 4497 |
. . . . 5
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10 | 8, 9 | ax-mp 5 |
. . . 4
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11 | 10 | rnmpt 5099 |
. . 3
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12 | eqid 2462 |
. . . 4
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13 | 12 | rnmpt 5099 |
. . 3
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14 | 6, 11, 13 | 3eqtr4i 2494 |
. 2
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15 | restval 15374 |
. . . . 5
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16 | 15 | 3adant3 1034 |
. . . 4
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17 | 16 | oveq1d 6330 |
. . 3
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18 | ovex 6343 |
. . . . 5
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19 | 16, 18 | syl6eqelr 2549 |
. . . 4
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20 | simp3 1016 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | restval 15374 |
. . . 4
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22 | 19, 20, 21 | syl2anc 671 |
. . 3
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23 | 17, 22 | eqtrd 2496 |
. 2
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24 | simp1 1014 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | inex1g 4560 |
. . . 4
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26 | 25 | 3ad2ant2 1036 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | restval 15374 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 24, 26, 27 | syl2anc 671 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 14, 23, 28 | 3eqtr4a 2522 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pr 4653 ax-un 6610 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-rest 15370 |
This theorem is referenced by: restabs 20230 restin 20231 resstopn 20251 ressuss 21327 |
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