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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 2689 | . . . . 5 | |
2 | 1 | inex1 4062 | . . . 4 |
3 | ineq1 3270 | . . . . 5 | |
4 | inass 3286 | . . . . 5 | |
5 | 3, 4 | syl6eq 2188 | . . . 4 |
6 | 2, 5 | abrexco 5660 | . . 3 |
7 | eqid 2139 | . . . . . 6 | |
8 | 7 | rnmpt 4787 | . . . . 5 |
9 | mpteq1 4012 | . . . . 5 | |
10 | 8, 9 | ax-mp 5 | . . . 4 |
11 | 10 | rnmpt 4787 | . . 3 |
12 | eqid 2139 | . . . 4 | |
13 | 12 | rnmpt 4787 | . . 3 |
14 | 6, 11, 13 | 3eqtr4i 2170 | . 2 |
15 | restval 12126 | . . . . 5 ↾t | |
16 | 15 | 3adant3 1001 | . . . 4 ↾t |
17 | 16 | oveq1d 5789 | . . 3 ↾t ↾t ↾t |
18 | restfn 12124 | . . . . . 6 ↾t | |
19 | simp1 981 | . . . . . . 7 | |
20 | 19 | elexd 2699 | . . . . . 6 |
21 | simp2 982 | . . . . . . 7 | |
22 | 21 | elexd 2699 | . . . . . 6 |
23 | fnovex 5804 | . . . . . 6 ↾t ↾t | |
24 | 18, 20, 22, 23 | mp3an2i 1320 | . . . . 5 ↾t |
25 | 16, 24 | eqeltrrd 2217 | . . . 4 |
26 | simp3 983 | . . . 4 | |
27 | restval 12126 | . . . 4 ↾t | |
28 | 25, 26, 27 | syl2anc 408 | . . 3 ↾t |
29 | 17, 28 | eqtrd 2172 | . 2 ↾t ↾t |
30 | inex1g 4064 | . . . 4 | |
31 | 30 | 3ad2ant2 1003 | . . 3 |
32 | restval 12126 | . . 3 ↾t | |
33 | 19, 31, 32 | syl2anc 408 | . 2 ↾t |
34 | 14, 29, 33 | 3eqtr4a 2198 | 1 ↾t ↾t ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 962 wceq 1331 wcel 1480 cab 2125 wrex 2417 cvv 2686 cin 3070 cmpt 3989 cxp 4537 crn 4540 wfn 5118 (class class class)co 5774 ↾t crest 12120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-rest 12122 |
This theorem is referenced by: restabs 12344 restin 12345 |
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