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Mirrors > Home > MPE Home > Th. List > f1dom3fv3dif | Structured version Visualization version Unicode version |
Description: The function values for a 1-1 function from a set with three different elements are different. (Contributed by AV, 20-Mar-2019.) |
Ref | Expression |
---|---|
f1dom3fv3dif.v |
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f1dom3fv3dif.n |
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f1dom3fv3dif.f |
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Ref | Expression |
---|---|
f1dom3fv3dif |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1dom3fv3dif.n |
. . . 4
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2 | 1 | simp1d 1042 |
. . 3
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3 | f1dom3fv3dif.f |
. . . . 5
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4 | eqidd 2472 |
. . . . . . 7
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5 | 4 | 3mix1d 1205 |
. . . . . 6
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6 | f1dom3fv3dif.v |
. . . . . . . 8
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7 | 6 | simp1d 1042 |
. . . . . . 7
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8 | eltpg 4005 |
. . . . . . 7
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9 | 7, 8 | syl 17 |
. . . . . 6
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10 | 5, 9 | mpbird 240 |
. . . . 5
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11 | eqidd 2472 |
. . . . . . 7
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12 | 11 | 3mix2d 1206 |
. . . . . 6
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13 | 6 | simp2d 1043 |
. . . . . . 7
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14 | eltpg 4005 |
. . . . . . 7
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15 | 13, 14 | syl 17 |
. . . . . 6
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16 | 12, 15 | mpbird 240 |
. . . . 5
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17 | f1fveq 6181 |
. . . . 5
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18 | 3, 10, 16, 17 | syl12anc 1290 |
. . . 4
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19 | 18 | necon3bid 2687 |
. . 3
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20 | 2, 19 | mpbird 240 |
. 2
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21 | 1 | simp2d 1043 |
. . 3
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22 | 6 | simp3d 1044 |
. . . . . 6
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23 | tpid3g 4078 |
. . . . . 6
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24 | 22, 23 | syl 17 |
. . . . 5
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25 | f1fveq 6181 |
. . . . 5
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26 | 3, 10, 24, 25 | syl12anc 1290 |
. . . 4
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27 | 26 | necon3bid 2687 |
. . 3
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28 | 21, 27 | mpbird 240 |
. 2
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29 | 1 | simp3d 1044 |
. . 3
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30 | f1fveq 6181 |
. . . . 5
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31 | 3, 16, 24, 30 | syl12anc 1290 |
. . . 4
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32 | 31 | necon3bid 2687 |
. . 3
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33 | 29, 32 | mpbird 240 |
. 2
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34 | 20, 28, 33 | 3jca 1210 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fv 5597 |
This theorem is referenced by: f1dom3el3dif 6187 |
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