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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 | ↾t ↾t ↾t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2738 | . . . . 5 | |
2 | 1 | inex1 4132 | . . . 4 |
3 | ineq1 3327 | . . . . 5 | |
4 | inass 3343 | . . . . 5 | |
5 | 3, 4 | eqtrdi 2224 | . . . 4 |
6 | 2, 5 | abrexco 5750 | . . 3 |
7 | eqid 2175 | . . . . . 6 | |
8 | 7 | rnmpt 4868 | . . . . 5 |
9 | mpteq1 4082 | . . . . 5 | |
10 | 8, 9 | ax-mp 5 | . . . 4 |
11 | 10 | rnmpt 4868 | . . 3 |
12 | eqid 2175 | . . . 4 | |
13 | 12 | rnmpt 4868 | . . 3 |
14 | 6, 11, 13 | 3eqtr4i 2206 | . 2 |
15 | restval 12614 | . . . . 5 ↾t | |
16 | 15 | 3adant3 1017 | . . . 4 ↾t |
17 | 16 | oveq1d 5880 | . . 3 ↾t ↾t ↾t |
18 | restfn 12612 | . . . . . 6 ↾t | |
19 | simp1 997 | . . . . . . 7 | |
20 | 19 | elexd 2748 | . . . . . 6 |
21 | simp2 998 | . . . . . . 7 | |
22 | 21 | elexd 2748 | . . . . . 6 |
23 | fnovex 5898 | . . . . . 6 ↾t ↾t | |
24 | 18, 20, 22, 23 | mp3an2i 1342 | . . . . 5 ↾t |
25 | 16, 24 | eqeltrrd 2253 | . . . 4 |
26 | simp3 999 | . . . 4 | |
27 | restval 12614 | . . . 4 ↾t | |
28 | 25, 26, 27 | syl2anc 411 | . . 3 ↾t |
29 | 17, 28 | eqtrd 2208 | . 2 ↾t ↾t |
30 | inex1g 4134 | . . . 4 | |
31 | 30 | 3ad2ant2 1019 | . . 3 |
32 | restval 12614 | . . 3 ↾t | |
33 | 19, 31, 32 | syl2anc 411 | . 2 ↾t |
34 | 14, 29, 33 | 3eqtr4a 2234 | 1 ↾t ↾t ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 978 wceq 1353 wcel 2146 cab 2161 wrex 2454 cvv 2735 cin 3126 cmpt 4059 cxp 4618 crn 4621 wfn 5203 (class class class)co 5865 ↾t crest 12608 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-rest 12610 |
This theorem is referenced by: restabs 13226 restin 13227 |
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