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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 2543 | . . . . 5 |
3 | eqid 2170 | . . . . . 6 | |
4 | 3 | fnmpt 5324 | . . . . 5 |
5 | 2, 4 | syl 14 | . . . 4 |
6 | offval2.4 | . . . . 5 | |
7 | 6 | fneq1d 5288 | . . . 4 |
8 | 5, 7 | mpbird 166 | . . 3 |
9 | offval2.3 | . . . . . 6 | |
10 | 9 | ralrimiva 2543 | . . . . 5 |
11 | eqid 2170 | . . . . . 6 | |
12 | 11 | fnmpt 5324 | . . . . 5 |
13 | 10, 12 | syl 14 | . . . 4 |
14 | offval2.5 | . . . . 5 | |
15 | 14 | fneq1d 5288 | . . . 4 |
16 | 13, 15 | mpbird 166 | . . 3 |
17 | offval2.1 | . . 3 | |
18 | inidm 3336 | . . 3 | |
19 | 6 | adantr 274 | . . . 4 |
20 | 19 | fveq1d 5498 | . . 3 |
21 | 14 | adantr 274 | . . . 4 |
22 | 21 | fveq1d 5498 | . . 3 |
23 | 8, 16, 17, 17, 18, 20, 22 | offval 6068 | . 2 |
24 | nffvmpt1 5507 | . . . . 5 | |
25 | nfcv 2312 | . . . . 5 | |
26 | nffvmpt1 5507 | . . . . 5 | |
27 | 24, 25, 26 | nfov 5883 | . . . 4 |
28 | nfcv 2312 | . . . 4 | |
29 | fveq2 5496 | . . . . 5 | |
30 | fveq2 5496 | . . . . 5 | |
31 | 29, 30 | oveq12d 5871 | . . . 4 |
32 | 27, 28, 31 | cbvmpt 4084 | . . 3 |
33 | simpr 109 | . . . . . 6 | |
34 | 3 | fvmpt2 5579 | . . . . . 6 |
35 | 33, 1, 34 | syl2anc 409 | . . . . 5 |
36 | 11 | fvmpt2 5579 | . . . . . 6 |
37 | 33, 9, 36 | syl2anc 409 | . . . . 5 |
38 | 35, 37 | oveq12d 5871 | . . . 4 |
39 | 38 | mpteq2dva 4079 | . . 3 |
40 | 32, 39 | eqtrid 2215 | . 2 |
41 | 23, 40 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wral 2448 cmpt 4050 wfn 5193 cfv 5198 (class class class)co 5853 cof 6059 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-of 6061 |
This theorem is referenced by: ofc12 6081 caofinvl 6083 caofcom 6084 dvimulf 13464 dvexp 13469 dvmptaddx 13475 dvmptmulx 13476 dvef 13482 |
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