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Mirrors > Home > ILE Home > Th. List > offres | Unicode version |
Description: Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
offres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 3158 | . . . . . 6 | |
2 | 1 | sseli 2941 | . . . . 5 |
3 | fvres 5198 | . . . . . 6 | |
4 | fvres 5198 | . . . . . 6 | |
5 | 3, 4 | oveq12d 5530 | . . . . 5 |
6 | 2, 5 | syl 14 | . . . 4 |
7 | 6 | mpteq2ia 3843 | . . 3 |
8 | inindi 3154 | . . . . 5 | |
9 | incom 3129 | . . . . 5 | |
10 | dmres 4632 | . . . . . 6 | |
11 | dmres 4632 | . . . . . 6 | |
12 | 10, 11 | ineq12i 3136 | . . . . 5 |
13 | 8, 9, 12 | 3eqtr4ri 2071 | . . . 4 |
14 | eqid 2040 | . . . 4 | |
15 | 13, 14 | mpteq12i 3845 | . . 3 |
16 | resmpt3 4657 | . . 3 | |
17 | 7, 15, 16 | 3eqtr4ri 2071 | . 2 |
18 | offval3 5761 | . . 3 | |
19 | 18 | reseq1d 4611 | . 2 |
20 | resexg 4650 | . . 3 | |
21 | resexg 4650 | . . 3 | |
22 | offval3 5761 | . . 3 | |
23 | 20, 21, 22 | syl2an 273 | . 2 |
24 | 17, 19, 23 | 3eqtr4a 2098 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cvv 2557 cin 2916 cmpt 3818 cdm 4345 cres 4347 cfv 4902 (class class class)co 5512 cof 5710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-of 5712 |
This theorem is referenced by: (None) |
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