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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 2663 | . . . . 5 | |
2 | 1 | inex1 4032 | . . . 4 |
3 | ineq1 3240 | . . . . 5 | |
4 | inass 3256 | . . . . 5 | |
5 | 3, 4 | syl6eq 2166 | . . . 4 |
6 | 2, 5 | abrexco 5628 | . . 3 |
7 | eqid 2117 | . . . . . 6 | |
8 | 7 | rnmpt 4757 | . . . . 5 |
9 | mpteq1 3982 | . . . . 5 | |
10 | 8, 9 | ax-mp 5 | . . . 4 |
11 | 10 | rnmpt 4757 | . . 3 |
12 | eqid 2117 | . . . 4 | |
13 | 12 | rnmpt 4757 | . . 3 |
14 | 6, 11, 13 | 3eqtr4i 2148 | . 2 |
15 | restval 12053 | . . . . 5 ↾t | |
16 | 15 | 3adant3 986 | . . . 4 ↾t |
17 | 16 | oveq1d 5757 | . . 3 ↾t ↾t ↾t |
18 | restfn 12051 | . . . . . 6 ↾t | |
19 | simp1 966 | . . . . . . 7 | |
20 | 19 | elexd 2673 | . . . . . 6 |
21 | simp2 967 | . . . . . . 7 | |
22 | 21 | elexd 2673 | . . . . . 6 |
23 | fnovex 5772 | . . . . . 6 ↾t ↾t | |
24 | 18, 20, 22, 23 | mp3an2i 1305 | . . . . 5 ↾t |
25 | 16, 24 | eqeltrrd 2195 | . . . 4 |
26 | simp3 968 | . . . 4 | |
27 | restval 12053 | . . . 4 ↾t | |
28 | 25, 26, 27 | syl2anc 408 | . . 3 ↾t |
29 | 17, 28 | eqtrd 2150 | . 2 ↾t ↾t |
30 | inex1g 4034 | . . . 4 | |
31 | 30 | 3ad2ant2 988 | . . 3 |
32 | restval 12053 | . . 3 ↾t | |
33 | 19, 31, 32 | syl2anc 408 | . 2 ↾t |
34 | 14, 29, 33 | 3eqtr4a 2176 | 1 ↾t ↾t ↾t |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 947 wceq 1316 wcel 1465 cab 2103 wrex 2394 cvv 2660 cin 3040 cmpt 3959 cxp 4507 crn 4510 wfn 5088 (class class class)co 5742 ↾t crest 12047 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-rest 12049 |
This theorem is referenced by: restabs 12271 restin 12272 |
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