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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 5145 |
. . 3
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2 | 1 | 3ad2ant1 967 |
. 2
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3 | fnfun 5145 |
. . 3
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4 | 3 | 3ad2ant2 968 |
. 2
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5 | fndm 5147 |
. . . . . . 7
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6 | fndm 5147 |
. . . . . . 7
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7 | 5, 6 | ineqan12d 3218 |
. . . . . 6
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8 | 7 | eqeq1d 2103 |
. . . . 5
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9 | 8 | biimprd 157 |
. . . 4
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10 | 9 | adantrd 274 |
. . 3
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11 | 10 | 3impia 1143 |
. 2
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12 | simp3r 975 |
. . 3
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13 | 5 | eleq2d 2164 |
. . . 4
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14 | 13 | 3ad2ant1 967 |
. . 3
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15 | 12, 14 | mpbird 166 |
. 2
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16 | funun 5092 |
. . . . . . 7
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17 | ssun1 3178 |
. . . . . . . . 9
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18 | dmss 4666 |
. . . . . . . . 9
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19 | 17, 18 | ax-mp 7 |
. . . . . . . 8
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20 | 19 | sseli 3035 |
. . . . . . 7
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21 | 16, 20 | anim12i 332 |
. . . . . 6
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22 | 21 | anasss 392 |
. . . . 5
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23 | 22 | 3impa 1141 |
. . . 4
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24 | funfvdm 5402 |
. . . 4
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25 | 23, 24 | syl 14 |
. . 3
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26 | imaundir 4878 |
. . . . . 6
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27 | 26 | a1i 9 |
. . . . 5
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28 | 27 | unieqd 3686 |
. . . 4
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29 | disjel 3356 |
. . . . . . . . 9
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30 | ndmima 4842 |
. . . . . . . . 9
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31 | 29, 30 | syl 14 |
. . . . . . . 8
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32 | 31 | 3ad2ant3 969 |
. . . . . . 7
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33 | 32 | uneq2d 3169 |
. . . . . 6
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34 | un0 3335 |
. . . . . 6
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35 | 33, 34 | syl6eq 2143 |
. . . . 5
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36 | 35 | unieqd 3686 |
. . . 4
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37 | 28, 36 | eqtrd 2127 |
. . 3
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38 | funfvdm 5402 |
. . . . . 6
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39 | 38 | eqcomd 2100 |
. . . . 5
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40 | 39 | adantrl 463 |
. . . 4
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41 | 40 | 3adant2 965 |
. . 3
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42 | 25, 37, 41 | 3eqtrd 2131 |
. 2
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43 | 2, 4, 11, 15, 42 | syl112anc 1185 |
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-sep 3978 ax-pow 4030 ax-pr 4060 |
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-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 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-fv 5057 |
This theorem is referenced by: fvun2 5406 caseinl 6862 |
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