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Mirrors > Home > ILE Home > Th. List > mpt2mptsx | Unicode version |
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
mpt2mptsx |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 |
. . . . . 6
![]() ![]() ![]() ![]() | |
2 | vex 2554 |
. . . . . 6
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3 | 1, 2 | op1std 5717 |
. . . . 5
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4 | 3 | csbeq1d 2852 |
. . . 4
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5 | 1, 2 | op2ndd 5718 |
. . . . . 6
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6 | 5 | csbeq1d 2852 |
. . . . 5
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7 | 6 | csbeq2dv 2869 |
. . . 4
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8 | 4, 7 | eqtrd 2069 |
. . 3
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9 | 8 | mpt2mptx 5537 |
. 2
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10 | nfcv 2175 |
. . . 4
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11 | nfcv 2175 |
. . . . 5
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12 | nfcsb1v 2876 |
. . . . 5
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13 | 11, 12 | nfxp 4314 |
. . . 4
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14 | sneq 3378 |
. . . . 5
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15 | csbeq1a 2854 |
. . . . 5
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16 | 14, 15 | xpeq12d 4313 |
. . . 4
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17 | 10, 13, 16 | cbviun 3685 |
. . 3
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18 | mpteq1 3832 |
. . 3
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19 | 17, 18 | ax-mp 7 |
. 2
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20 | nfcv 2175 |
. . 3
![]() ![]() ![]() ![]() | |
21 | nfcv 2175 |
. . 3
![]() ![]() ![]() ![]() | |
22 | nfcv 2175 |
. . 3
![]() ![]() ![]() ![]() | |
23 | nfcsb1v 2876 |
. . 3
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24 | nfcv 2175 |
. . . 4
![]() ![]() ![]() ![]() | |
25 | nfcsb1v 2876 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 24, 25 | nfcsb 2878 |
. . 3
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27 | csbeq1a 2854 |
. . . 4
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28 | csbeq1a 2854 |
. . . 4
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29 | 27, 28 | sylan9eqr 2091 |
. . 3
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30 | 20, 12, 21, 22, 23, 26, 15, 29 | cbvmpt2x 5524 |
. 2
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31 | 9, 19, 30 | 3eqtr4ri 2068 |
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 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fv 4853 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 |
This theorem is referenced by: mpt2mpts 5766 mpt2fvex 5771 |
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