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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 2505 | . . . . 5 |
3 | eqid 2139 | . . . . . 6 | |
4 | 3 | fnmpt 5249 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | offval2.4 | . . . . 5 | |
7 | 6 | fneq1d 5213 | . . . 4 |
8 | 5, 7 | mpbird 166 | . . 3 |
9 | offval2.3 | . . . . . 6 | |
10 | 9 | ralrimiva 2505 | . . . . 5 |
11 | eqid 2139 | . . . . . 6 | |
12 | 11 | fnmpt 5249 | . . . . 5 |
13 | 10, 12 | syl 14 | . . . 4 |
14 | offval2.5 | . . . . 5 | |
15 | 14 | fneq1d 5213 | . . . 4 |
16 | 13, 15 | mpbird 166 | . . 3 |
17 | offval2.1 | . . 3 | |
18 | inidm 3285 | . . 3 | |
19 | 6 | adantr 274 | . . . 4 |
20 | 19 | fveq1d 5423 | . . 3 |
21 | 14 | adantr 274 | . . . 4 |
22 | 21 | fveq1d 5423 | . . 3 |
23 | 8, 16, 17, 17, 18, 20, 22 | offval 5989 | . 2 |
24 | nffvmpt1 5432 | . . . . 5 | |
25 | nfcv 2281 | . . . . 5 | |
26 | nffvmpt1 5432 | . . . . 5 | |
27 | 24, 25, 26 | nfov 5801 | . . . 4 |
28 | nfcv 2281 | . . . 4 | |
29 | fveq2 5421 | . . . . 5 | |
30 | fveq2 5421 | . . . . 5 | |
31 | 29, 30 | oveq12d 5792 | . . . 4 |
32 | 27, 28, 31 | cbvmpt 4023 | . . 3 |
33 | simpr 109 | . . . . . 6 | |
34 | 3 | fvmpt2 5504 | . . . . . 6 |
35 | 33, 1, 34 | syl2anc 408 | . . . . 5 |
36 | 11 | fvmpt2 5504 | . . . . . 6 |
37 | 33, 9, 36 | syl2anc 408 | . . . . 5 |
38 | 35, 37 | oveq12d 5792 | . . . 4 |
39 | 38 | mpteq2dva 4018 | . . 3 |
40 | 32, 39 | syl5eq 2184 | . 2 |
41 | 23, 40 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 cmpt 3989 wfn 5118 cfv 5123 (class class class)co 5774 cof 5980 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-of 5982 |
This theorem is referenced by: ofc12 6002 caofinvl 6004 caofcom 6005 dvimulf 12839 dvexp 12844 dvmptaddx 12850 dvmptmulx 12851 dvef 12856 |
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