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Mirrors > Home > ILE Home > Th. List > ofrfval | Unicode version |
Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-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 |
---|---|
ofrfval |
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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 5650 |
. . . 4
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4 | 1, 2, 3 | syl2anc 409 |
. . 3
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5 | offval.2 |
. . . 4
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6 | offval.4 |
. . . 4
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7 | fnex 5650 |
. . . 4
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8 | 5, 6, 7 | syl2anc 409 |
. . 3
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9 | dmeq 4747 |
. . . . . 6
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10 | dmeq 4747 |
. . . . . 6
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11 | 9, 10 | ineqan12d 3284 |
. . . . 5
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12 | fveq1 5428 |
. . . . . 6
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13 | fveq1 5428 |
. . . . . 6
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14 | 12, 13 | breqan12d 3953 |
. . . . 5
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15 | 11, 14 | raleqbidv 2641 |
. . . 4
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16 | df-ofr 5991 |
. . . 4
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17 | 15, 16 | brabga 4194 |
. . 3
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18 | 4, 8, 17 | syl2anc 409 |
. 2
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19 | fndm 5230 |
. . . . . 6
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20 | 1, 19 | syl 14 |
. . . . 5
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21 | fndm 5230 |
. . . . . 6
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22 | 5, 21 | syl 14 |
. . . . 5
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23 | 20, 22 | ineq12d 3283 |
. . . 4
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24 | offval.5 |
. . . 4
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25 | 23, 24 | eqtrdi 2189 |
. . 3
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26 | 25 | raleqdv 2635 |
. 2
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27 | inss1 3301 |
. . . . . . 7
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28 | 24, 27 | eqsstrri 3135 |
. . . . . 6
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29 | 28 | sseli 3098 |
. . . . 5
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30 | offval.6 |
. . . . 5
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31 | 29, 30 | sylan2 284 |
. . . 4
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32 | inss2 3302 |
. . . . . . 7
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33 | 24, 32 | eqsstrri 3135 |
. . . . . 6
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34 | 33 | sseli 3098 |
. . . . 5
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35 | offval.7 |
. . . . 5
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36 | 34, 35 | sylan2 284 |
. . . 4
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37 | 31, 36 | breq12d 3950 |
. . 3
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38 | 37 | ralbidva 2434 |
. 2
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39 | 18, 26, 38 | 3bitrd 213 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ofr 5991 |
This theorem is referenced by: ofrval 6000 ofrfval2 6006 caofref 6011 caofrss 6014 caoftrn 6015 |
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