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Mirrors > Home > ILE Home > Th. List > fvun1 | Unicode version |
Description: The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.) |
Ref | Expression |
---|---|
fvun1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 5335 |
. . 3
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2 | 1 | 3ad2ant1 1020 |
. 2
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3 | fnfun 5335 |
. . 3
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4 | 3 | 3ad2ant2 1021 |
. 2
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5 | fndm 5337 |
. . . . . . 7
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6 | fndm 5337 |
. . . . . . 7
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7 | 5, 6 | ineqan12d 3353 |
. . . . . 6
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8 | 7 | eqeq1d 2198 |
. . . . 5
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9 | 8 | biimprd 158 |
. . . 4
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10 | 9 | adantrd 279 |
. . 3
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11 | 10 | 3impia 1202 |
. 2
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12 | simp3r 1028 |
. . 3
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13 | 5 | eleq2d 2259 |
. . . 4
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14 | 13 | 3ad2ant1 1020 |
. . 3
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15 | 12, 14 | mpbird 167 |
. 2
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16 | funun 5282 |
. . . . . . 7
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17 | ssun1 3313 |
. . . . . . . . 9
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18 | dmss 4847 |
. . . . . . . . 9
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19 | 17, 18 | ax-mp 5 |
. . . . . . . 8
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20 | 19 | sseli 3166 |
. . . . . . 7
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21 | 16, 20 | anim12i 338 |
. . . . . 6
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22 | 21 | anasss 399 |
. . . . 5
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23 | 22 | 3impa 1196 |
. . . 4
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24 | funfvdm 5603 |
. . . 4
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25 | 23, 24 | syl 14 |
. . 3
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26 | imaundir 5063 |
. . . . . 6
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27 | 26 | a1i 9 |
. . . . 5
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28 | 27 | unieqd 3838 |
. . . 4
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29 | disjel 3492 |
. . . . . . . . 9
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30 | ndmima 5026 |
. . . . . . . . 9
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31 | 29, 30 | syl 14 |
. . . . . . . 8
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32 | 31 | 3ad2ant3 1022 |
. . . . . . 7
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33 | 32 | uneq2d 3304 |
. . . . . 6
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34 | un0 3471 |
. . . . . 6
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35 | 33, 34 | eqtrdi 2238 |
. . . . 5
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36 | 35 | unieqd 3838 |
. . . 4
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37 | 28, 36 | eqtrd 2222 |
. . 3
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38 | funfvdm 5603 |
. . . . . 6
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39 | 38 | eqcomd 2195 |
. . . . 5
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40 | 39 | adantrl 478 |
. . . 4
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41 | 40 | 3adant2 1018 |
. . 3
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42 | 25, 37, 41 | 3eqtrd 2226 |
. 2
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43 | 2, 4, 11, 15, 42 | syl112anc 1253 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-fv 5246 |
This theorem is referenced by: fvun2 5607 caseinl 7124 |
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