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Mirrors > Home > ILE Home > Th. List > funtpg | Unicode version |
Description: A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
Ref | Expression |
---|---|
funtpg |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 979 |
. . . 4
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2 | 3simpa 979 |
. . . 4
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3 | simp1 982 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | funprg 5181 |
. . . 4
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5 | 1, 2, 3, 4 | syl3an 1259 |
. . 3
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6 | simp13 1014 |
. . . 4
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7 | simp23 1017 |
. . . 4
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8 | funsng 5177 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 6, 7, 8 | syl2anc 409 |
. . 3
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10 | 2 | 3ad2ant2 1004 |
. . . . . 6
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11 | dmpropg 5019 |
. . . . . 6
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12 | 10, 11 | syl 14 |
. . . . 5
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13 | dmsnopg 5018 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 7, 13 | syl 14 |
. . . . 5
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15 | 12, 14 | ineq12d 3283 |
. . . 4
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16 | elpri 3555 |
. . . . . . . 8
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17 | nner 2313 |
. . . . . . . . . . . 12
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18 | 17 | eqcoms 2143 |
. . . . . . . . . . 11
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19 | 3mix2 1152 |
. . . . . . . . . . 11
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20 | 18, 19 | syl 14 |
. . . . . . . . . 10
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21 | nner 2313 |
. . . . . . . . . . . 12
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22 | 21 | eqcoms 2143 |
. . . . . . . . . . 11
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23 | 3mix3 1153 |
. . . . . . . . . . 11
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24 | 22, 23 | syl 14 |
. . . . . . . . . 10
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25 | 20, 24 | jaoi 706 |
. . . . . . . . 9
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26 | 3ianorr 1288 |
. . . . . . . . 9
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27 | 25, 26 | syl 14 |
. . . . . . . 8
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28 | 16, 27 | syl 14 |
. . . . . . 7
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29 | 28 | con2i 617 |
. . . . . 6
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30 | disjsn 3593 |
. . . . . 6
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31 | 29, 30 | sylibr 133 |
. . . . 5
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32 | 31 | 3ad2ant3 1005 |
. . . 4
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33 | 15, 32 | eqtrd 2173 |
. . 3
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34 | funun 5175 |
. . 3
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35 | 5, 9, 33, 34 | syl21anc 1216 |
. 2
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36 | df-tp 3540 |
. . 3
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37 | 36 | funeqi 5152 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 35, 37 | sylibr 133 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-tp 3540 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-fun 5133 |
This theorem is referenced by: fntpg 5187 |
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