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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 5558 |
. . . 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 5558 |
. . . 4
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8 | 5, 6, 7 | syl2anc 404 |
. . 3
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9 | fndm 5147 |
. . . . . . . 8
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10 | 1, 9 | syl 14 |
. . . . . . 7
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11 | fndm 5147 |
. . . . . . . 8
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12 | 5, 11 | syl 14 |
. . . . . . 7
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13 | 10, 12 | ineq12d 3217 |
. . . . . 6
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14 | offval.5 |
. . . . . 6
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15 | 13, 14 | syl6eq 2143 |
. . . . 5
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16 | 15 | mpteq1d 3945 |
. . . 4
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17 | inex1g 3996 |
. . . . . 6
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18 | 14, 17 | syl5eqelr 2182 |
. . . . 5
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19 | mptexg 5561 |
. . . . 5
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20 | 2, 18, 19 | 3syl 17 |
. . . 4
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21 | 16, 20 | eqeltrd 2171 |
. . 3
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22 | dmeq 4667 |
. . . . . 6
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23 | dmeq 4667 |
. . . . . 6
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24 | 22, 23 | ineqan12d 3218 |
. . . . 5
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25 | fveq1 5339 |
. . . . . 6
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26 | fveq1 5339 |
. . . . . 6
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27 | 25, 26 | oveqan12d 5709 |
. . . . 5
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28 | 24, 27 | mpteq12dv 3942 |
. . . 4
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29 | df-of 5894 |
. . . 4
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30 | 28, 29 | ovmpt2ga 5812 |
. . 3
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31 | 4, 8, 21, 30 | syl3anc 1181 |
. 2
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32 | 14 | eleq2i 2161 |
. . . . 5
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33 | elin 3198 |
. . . . 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 5708 |
. . . 4
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40 | 34, 39 | sylan2b 282 |
. . 3
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41 | 40 | mpteq2dva 3950 |
. 2
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42 | 31, 16, 41 | 3eqtrd 2131 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-setind 4381 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-of 5894 |
This theorem is referenced by: fnofval 5903 off 5906 ofres 5907 offval2 5908 suppssof1 5910 ofco 5911 offveqb 5912 |
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