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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 5737 |
. . . 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 5737 |
. . . 4
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8 | 5, 6, 7 | syl2anc 411 |
. . 3
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9 | dmeq 4826 |
. . . . . 6
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10 | dmeq 4826 |
. . . . . 6
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11 | 9, 10 | ineqan12d 3338 |
. . . . 5
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12 | fveq1 5513 |
. . . . . 6
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13 | fveq1 5513 |
. . . . . 6
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14 | 12, 13 | breqan12d 4018 |
. . . . 5
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15 | 11, 14 | raleqbidv 2684 |
. . . 4
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16 | df-ofr 6081 |
. . . 4
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17 | 15, 16 | brabga 4263 |
. . 3
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18 | 4, 8, 17 | syl2anc 411 |
. 2
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19 | fndm 5314 |
. . . . . 6
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20 | 1, 19 | syl 14 |
. . . . 5
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21 | fndm 5314 |
. . . . . 6
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22 | 5, 21 | syl 14 |
. . . . 5
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23 | 20, 22 | ineq12d 3337 |
. . . 4
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24 | offval.5 |
. . . 4
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25 | 23, 24 | eqtrdi 2226 |
. . 3
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26 | 25 | raleqdv 2678 |
. 2
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27 | inss1 3355 |
. . . . . . 7
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28 | 24, 27 | eqsstrri 3188 |
. . . . . 6
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29 | 28 | sseli 3151 |
. . . . 5
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30 | offval.6 |
. . . . 5
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31 | 29, 30 | sylan2 286 |
. . . 4
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32 | inss2 3356 |
. . . . . . 7
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33 | 24, 32 | eqsstrri 3188 |
. . . . . 6
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34 | 33 | sseli 3151 |
. . . . 5
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35 | offval.7 |
. . . . 5
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36 | 34, 35 | sylan2 286 |
. . . 4
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37 | 31, 36 | breq12d 4015 |
. . 3
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38 | 37 | ralbidva 2473 |
. 2
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39 | 18, 26, 38 | 3bitrd 214 |
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-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 4117 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-ofr 6081 |
This theorem is referenced by: ofrval 6090 ofrfval2 6096 caofref 6101 caofrss 6104 caoftrn 6105 |
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