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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 |
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offval.2 |
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offval.3 |
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offval.4 |
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offval.5 |
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offval.6 |
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offval.7 |
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Ref | Expression |
---|---|
offval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval.1 |
. . . 4
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2 | offval.3 |
. . . 4
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3 | fnex 5533 |
. . . 4
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4 | 1, 2, 3 | syl2anc 404 |
. . 3
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5 | offval.2 |
. . . 4
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6 | offval.4 |
. . . 4
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7 | fnex 5533 |
. . . 4
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8 | 5, 6, 7 | syl2anc 404 |
. . 3
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9 | fndm 5126 |
. . . . . . . 8
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10 | 1, 9 | syl 14 |
. . . . . . 7
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11 | fndm 5126 |
. . . . . . . 8
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12 | 5, 11 | syl 14 |
. . . . . . 7
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13 | 10, 12 | ineq12d 3203 |
. . . . . 6
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14 | offval.5 |
. . . . . 6
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15 | 13, 14 | syl6eq 2137 |
. . . . 5
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16 | 15 | mpteq1d 3929 |
. . . 4
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17 | inex1g 3981 |
. . . . . 6
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18 | 14, 17 | syl5eqelr 2176 |
. . . . 5
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19 | mptexg 5536 |
. . . . 5
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20 | 2, 18, 19 | 3syl 17 |
. . . 4
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21 | 16, 20 | eqeltrd 2165 |
. . 3
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22 | dmeq 4649 |
. . . . . 6
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23 | dmeq 4649 |
. . . . . 6
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24 | 22, 23 | ineqan12d 3204 |
. . . . 5
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25 | fveq1 5317 |
. . . . . 6
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26 | fveq1 5317 |
. . . . . 6
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27 | 25, 26 | oveqan12d 5685 |
. . . . 5
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28 | 24, 27 | mpteq12dv 3926 |
. . . 4
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29 | df-of 5870 |
. . . 4
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30 | 28, 29 | ovmpt2ga 5788 |
. . 3
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31 | 4, 8, 21, 30 | syl3anc 1175 |
. 2
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32 | 14 | eleq2i 2155 |
. . . . 5
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33 | elin 3184 |
. . . . 5
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34 | 32, 33 | bitr3i 185 |
. . . 4
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35 | offval.6 |
. . . . . 6
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36 | 35 | adantrr 464 |
. . . . 5
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37 | offval.7 |
. . . . . 6
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38 | 37 | adantrl 463 |
. . . . 5
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39 | 36, 38 | oveq12d 5684 |
. . . 4
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40 | 34, 39 | sylan2b 282 |
. . 3
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41 | 40 | mpteq2dva 3934 |
. 2
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42 | 31, 16, 41 | 3eqtrd 2125 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-of 5870 |
This theorem is referenced by: fnofval 5879 off 5882 ofres 5883 offval2 5884 suppssof1 5886 ofco 5887 offveqb 5888 |
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