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Mirrors > Home > ILE Home > Th. List > offval2 | Unicode version |
Description: The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
offval2.1 | |
offval2.2 | |
offval2.3 | |
offval2.4 | |
offval2.5 |
Ref | Expression |
---|---|
offval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval2.2 | . . . . . 6 | |
2 | 1 | ralrimiva 2434 | . . . . 5 |
3 | eqid 2081 | . . . . . 6 | |
4 | 3 | fnmpt 5045 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | offval2.4 | . . . . 5 | |
7 | 6 | fneq1d 5009 | . . . 4 |
8 | 5, 7 | mpbird 165 | . . 3 |
9 | offval2.3 | . . . . . 6 | |
10 | 9 | ralrimiva 2434 | . . . . 5 |
11 | eqid 2081 | . . . . . 6 | |
12 | 11 | fnmpt 5045 | . . . . 5 |
13 | 10, 12 | syl 14 | . . . 4 |
14 | offval2.5 | . . . . 5 | |
15 | 14 | fneq1d 5009 | . . . 4 |
16 | 13, 15 | mpbird 165 | . . 3 |
17 | offval2.1 | . . 3 | |
18 | inidm 3175 | . . 3 | |
19 | 6 | adantr 270 | . . . 4 |
20 | 19 | fveq1d 5200 | . . 3 |
21 | 14 | adantr 270 | . . . 4 |
22 | 21 | fveq1d 5200 | . . 3 |
23 | 8, 16, 17, 17, 18, 20, 22 | offval 5739 | . 2 |
24 | nffvmpt1 5206 | . . . . 5 | |
25 | nfcv 2219 | . . . . 5 | |
26 | nffvmpt1 5206 | . . . . 5 | |
27 | 24, 25, 26 | nfov 5555 | . . . 4 |
28 | nfcv 2219 | . . . 4 | |
29 | fveq2 5198 | . . . . 5 | |
30 | fveq2 5198 | . . . . 5 | |
31 | 29, 30 | oveq12d 5550 | . . . 4 |
32 | 27, 28, 31 | cbvmpt 3872 | . . 3 |
33 | simpr 108 | . . . . . 6 | |
34 | 3 | fvmpt2 5275 | . . . . . 6 |
35 | 33, 1, 34 | syl2anc 403 | . . . . 5 |
36 | 11 | fvmpt2 5275 | . . . . . 6 |
37 | 33, 9, 36 | syl2anc 403 | . . . . 5 |
38 | 35, 37 | oveq12d 5550 | . . . 4 |
39 | 38 | mpteq2dva 3868 | . . 3 |
40 | 32, 39 | syl5eq 2125 | . 2 |
41 | 23, 40 | eqtrd 2113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 wral 2348 cmpt 3839 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: ofc12 5751 caofinvl 5753 caofcom 5754 |
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