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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 5740 |
. . . 4
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4 | 1, 2, 3 | syl2anc 411 |
. . 3
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5 | offval.2 |
. . . 4
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6 | offval.4 |
. . . 4
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7 | fnex 5740 |
. . . 4
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8 | 5, 6, 7 | syl2anc 411 |
. . 3
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9 | fndm 5317 |
. . . . . . . 8
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10 | 1, 9 | syl 14 |
. . . . . . 7
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11 | fndm 5317 |
. . . . . . . 8
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12 | 5, 11 | syl 14 |
. . . . . . 7
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13 | 10, 12 | ineq12d 3339 |
. . . . . 6
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14 | offval.5 |
. . . . . 6
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15 | 13, 14 | eqtrdi 2226 |
. . . . 5
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16 | 15 | mpteq1d 4090 |
. . . 4
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17 | inex1g 4141 |
. . . . . 6
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18 | 14, 17 | eqeltrrid 2265 |
. . . . 5
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19 | mptexg 5743 |
. . . . 5
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20 | 2, 18, 19 | 3syl 17 |
. . . 4
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21 | 16, 20 | eqeltrd 2254 |
. . 3
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22 | dmeq 4829 |
. . . . . 6
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23 | dmeq 4829 |
. . . . . 6
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24 | 22, 23 | ineqan12d 3340 |
. . . . 5
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25 | fveq1 5516 |
. . . . . 6
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26 | fveq1 5516 |
. . . . . 6
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27 | 25, 26 | oveqan12d 5896 |
. . . . 5
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28 | 24, 27 | mpteq12dv 4087 |
. . . 4
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29 | df-of 6085 |
. . . 4
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30 | 28, 29 | ovmpoga 6006 |
. . 3
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31 | 4, 8, 21, 30 | syl3anc 1238 |
. 2
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32 | 14 | eleq2i 2244 |
. . . . 5
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33 | elin 3320 |
. . . . 5
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34 | 32, 33 | bitr3i 186 |
. . . 4
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35 | offval.6 |
. . . . . 6
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36 | 35 | adantrr 479 |
. . . . 5
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37 | offval.7 |
. . . . . 6
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38 | 37 | adantrl 478 |
. . . . 5
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39 | 36, 38 | oveq12d 5895 |
. . . 4
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40 | 34, 39 | sylan2b 287 |
. . 3
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41 | 40 | mpteq2dva 4095 |
. 2
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42 | 31, 16, 41 | 3eqtrd 2214 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-setind 4538 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-of 6085 |
This theorem is referenced by: ofvalg 6094 off 6097 ofres 6099 offval2 6100 suppssof1 6102 ofco 6103 offveqb 6104 |
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