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Mirrors > Home > ILE Home > Th. List > offval | Unicode version |
Description: Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
offval.1 | |
offval.2 | |
offval.3 | |
offval.4 | |
offval.5 | |
offval.6 | |
offval.7 |
Ref | Expression |
---|---|
offval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval.1 | . . . 4 | |
2 | offval.3 | . . . 4 | |
3 | fnex 5404 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 403 | . . 3 |
5 | offval.2 | . . . 4 | |
6 | offval.4 | . . . 4 | |
7 | fnex 5404 | . . . 4 | |
8 | 5, 6, 7 | syl2anc 403 | . . 3 |
9 | fndm 5018 | . . . . . . . 8 | |
10 | 1, 9 | syl 14 | . . . . . . 7 |
11 | fndm 5018 | . . . . . . . 8 | |
12 | 5, 11 | syl 14 | . . . . . . 7 |
13 | 10, 12 | ineq12d 3168 | . . . . . 6 |
14 | offval.5 | . . . . . 6 | |
15 | 13, 14 | syl6eq 2129 | . . . . 5 |
16 | 15 | mpteq1d 3863 | . . . 4 |
17 | inex1g 3914 | . . . . . 6 | |
18 | 14, 17 | syl5eqelr 2166 | . . . . 5 |
19 | mptexg 5407 | . . . . 5 | |
20 | 2, 18, 19 | 3syl 17 | . . . 4 |
21 | 16, 20 | eqeltrd 2155 | . . 3 |
22 | dmeq 4553 | . . . . . 6 | |
23 | dmeq 4553 | . . . . . 6 | |
24 | 22, 23 | ineqan12d 3169 | . . . . 5 |
25 | fveq1 5197 | . . . . . 6 | |
26 | fveq1 5197 | . . . . . 6 | |
27 | 25, 26 | oveqan12d 5551 | . . . . 5 |
28 | 24, 27 | mpteq12dv 3860 | . . . 4 |
29 | df-of 5732 | . . . 4 | |
30 | 28, 29 | ovmpt2ga 5650 | . . 3 |
31 | 4, 8, 21, 30 | syl3anc 1169 | . 2 |
32 | 14 | eleq2i 2145 | . . . . 5 |
33 | elin 3155 | . . . . 5 | |
34 | 32, 33 | bitr3i 184 | . . . 4 |
35 | offval.6 | . . . . . 6 | |
36 | 35 | adantrr 462 | . . . . 5 |
37 | offval.7 | . . . . . 6 | |
38 | 37 | adantrl 461 | . . . . 5 |
39 | 36, 38 | oveq12d 5550 | . . . 4 |
40 | 34, 39 | sylan2b 281 | . . 3 |
41 | 40 | mpteq2dva 3868 | . 2 |
42 | 31, 16, 41 | 3eqtrd 2117 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cvv 2601 cin 2972 cmpt 3839 cdm 4363 wfn 4917 cfv 4922 (class class class)co 5532 cof 5730 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-of 5732 |
This theorem is referenced by: fnofval 5741 off 5744 ofres 5745 offval2 5746 suppssof1 5748 ofco 5749 offveqb 5750 |
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