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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 2530 | . . . . 5 |
3 | eqid 2157 | . . . . . 6 | |
4 | 3 | fnmpt 5298 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | offval2.4 | . . . . 5 | |
7 | 6 | fneq1d 5262 | . . . 4 |
8 | 5, 7 | mpbird 166 | . . 3 |
9 | offval2.3 | . . . . . 6 | |
10 | 9 | ralrimiva 2530 | . . . . 5 |
11 | eqid 2157 | . . . . . 6 | |
12 | 11 | fnmpt 5298 | . . . . 5 |
13 | 10, 12 | syl 14 | . . . 4 |
14 | offval2.5 | . . . . 5 | |
15 | 14 | fneq1d 5262 | . . . 4 |
16 | 13, 15 | mpbird 166 | . . 3 |
17 | offval2.1 | . . 3 | |
18 | inidm 3317 | . . 3 | |
19 | 6 | adantr 274 | . . . 4 |
20 | 19 | fveq1d 5472 | . . 3 |
21 | 14 | adantr 274 | . . . 4 |
22 | 21 | fveq1d 5472 | . . 3 |
23 | 8, 16, 17, 17, 18, 20, 22 | offval 6041 | . 2 |
24 | nffvmpt1 5481 | . . . . 5 | |
25 | nfcv 2299 | . . . . 5 | |
26 | nffvmpt1 5481 | . . . . 5 | |
27 | 24, 25, 26 | nfov 5853 | . . . 4 |
28 | nfcv 2299 | . . . 4 | |
29 | fveq2 5470 | . . . . 5 | |
30 | fveq2 5470 | . . . . 5 | |
31 | 29, 30 | oveq12d 5844 | . . . 4 |
32 | 27, 28, 31 | cbvmpt 4061 | . . 3 |
33 | simpr 109 | . . . . . 6 | |
34 | 3 | fvmpt2 5553 | . . . . . 6 |
35 | 33, 1, 34 | syl2anc 409 | . . . . 5 |
36 | 11 | fvmpt2 5553 | . . . . . 6 |
37 | 33, 9, 36 | syl2anc 409 | . . . . 5 |
38 | 35, 37 | oveq12d 5844 | . . . 4 |
39 | 38 | mpteq2dva 4056 | . . 3 |
40 | 32, 39 | syl5eq 2202 | . 2 |
41 | 23, 40 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wral 2435 cmpt 4027 wfn 5167 cfv 5172 (class class class)co 5826 cof 6032 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-setind 4498 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-of 6034 |
This theorem is referenced by: ofc12 6054 caofinvl 6056 caofcom 6057 dvimulf 13140 dvexp 13145 dvmptaddx 13151 dvmptmulx 13152 dvef 13158 |
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