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Mirrors > Home > ILE Home > Th. List > offval2 | Unicode version |
Description: The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
offval2.1 |
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offval2.2 |
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offval2.3 |
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offval2.4 |
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offval2.5 |
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Ref | Expression |
---|---|
offval2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval2.2 |
. . . . . 6
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2 | 1 | ralrimiva 2386 |
. . . . 5
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3 | eqid 2037 |
. . . . . 6
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4 | 3 | fnmpt 4968 |
. . . . 5
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5 | 2, 4 | syl 14 |
. . . 4
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6 | offval2.4 |
. . . . 5
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7 | 6 | fneq1d 4932 |
. . . 4
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8 | 5, 7 | mpbird 156 |
. . 3
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9 | offval2.3 |
. . . . . 6
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10 | 9 | ralrimiva 2386 |
. . . . 5
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11 | eqid 2037 |
. . . . . 6
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12 | 11 | fnmpt 4968 |
. . . . 5
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13 | 10, 12 | syl 14 |
. . . 4
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14 | offval2.5 |
. . . . 5
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15 | 14 | fneq1d 4932 |
. . . 4
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16 | 13, 15 | mpbird 156 |
. . 3
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17 | offval2.1 |
. . 3
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18 | inidm 3140 |
. . 3
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19 | 6 | adantr 261 |
. . . 4
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20 | 19 | fveq1d 5123 |
. . 3
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21 | 14 | adantr 261 |
. . . 4
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22 | 21 | fveq1d 5123 |
. . 3
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23 | 8, 16, 17, 17, 18, 20, 22 | offval 5661 |
. 2
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24 | nffvmpt1 5129 |
. . . . 5
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25 | nfcv 2175 |
. . . . 5
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26 | nffvmpt1 5129 |
. . . . 5
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27 | 24, 25, 26 | nfov 5478 |
. . . 4
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28 | nfcv 2175 |
. . . 4
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29 | fveq2 5121 |
. . . . 5
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30 | fveq2 5121 |
. . . . 5
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31 | 29, 30 | oveq12d 5473 |
. . . 4
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32 | 27, 28, 31 | cbvmpt 3842 |
. . 3
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33 | simpr 103 |
. . . . . 6
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34 | 3 | fvmpt2 5197 |
. . . . . 6
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35 | 33, 1, 34 | syl2anc 391 |
. . . . 5
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36 | 11 | fvmpt2 5197 |
. . . . . 6
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37 | 33, 9, 36 | syl2anc 391 |
. . . . 5
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38 | 35, 37 | oveq12d 5473 |
. . . 4
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39 | 38 | mpteq2dva 3838 |
. . 3
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40 | 32, 39 | syl5eq 2081 |
. 2
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41 | 23, 40 | eqtrd 2069 |
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 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-of 5654 |
This theorem is referenced by: ofc12 5673 caofinvl 5675 caofcom 5676 |
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