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Mirrors > Home > ILE Home > Th. List > fmptpr | Unicode version |
Description: Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Ref | Expression |
---|---|
fmptpr.1 |
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fmptpr.2 |
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fmptpr.3 |
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fmptpr.4 |
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fmptpr.5 |
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fmptpr.6 |
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Ref | Expression |
---|---|
fmptpr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3617 |
. . 3
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2 | 1 | a1i 9 |
. 2
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3 | mpt0 5365 |
. . . . . 6
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4 | 3 | uneq1i 3300 |
. . . . 5
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5 | uncom 3294 |
. . . . 5
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6 | un0 3471 |
. . . . 5
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7 | 4, 5, 6 | 3eqtri 2214 |
. . . 4
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8 | fmptpr.1 |
. . . . . 6
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9 | elex 2763 |
. . . . . 6
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10 | 8, 9 | syl 14 |
. . . . 5
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11 | fmptpr.3 |
. . . . . 6
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12 | elex 2763 |
. . . . . 6
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13 | 11, 12 | syl 14 |
. . . . 5
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14 | uncom 3294 |
. . . . . . 7
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15 | un0 3471 |
. . . . . . 7
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16 | 14, 15 | eqtr3i 2212 |
. . . . . 6
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17 | 16 | a1i 9 |
. . . . 5
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18 | fmptpr.5 |
. . . . 5
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19 | 10, 13, 17, 18 | fmptapd 5731 |
. . . 4
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20 | 7, 19 | eqtr3id 2236 |
. . 3
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21 | 20 | uneq1d 3303 |
. 2
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22 | fmptpr.2 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | elex 2763 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | syl 14 |
. . 3
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25 | fmptpr.4 |
. . . 4
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26 | elex 2763 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 25, 26 | syl 14 |
. . 3
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28 | df-pr 3617 |
. . . . 5
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29 | 28 | eqcomi 2193 |
. . . 4
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30 | 29 | a1i 9 |
. . 3
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31 | fmptpr.6 |
. . 3
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32 | 24, 27, 30, 31 | fmptapd 5731 |
. 2
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33 | 2, 21, 32 | 3eqtrd 2226 |
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-nul 4147 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-reu 2475 df-v 2754 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-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 |
This theorem is referenced by: (None) |
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