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Mirrors > Home > MPE Home > Th. List > mpt2mptsx | Structured version 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 3081 |
. . . . . 6
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2 | vex 3081 |
. . . . . 6
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3 | 1, 2 | op1std 6700 |
. . . . 5
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4 | 3 | csbeq1d 3405 |
. . . 4
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5 | 1, 2 | op2ndd 6701 |
. . . . . 6
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6 | 5 | csbeq1d 3405 |
. . . . 5
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7 | 6 | csbeq2dv 3798 |
. . . 4
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8 | 4, 7 | eqtrd 2495 |
. . 3
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9 | 8 | mpt2mptx 6294 |
. 2
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10 | nfcv 2616 |
. . . 4
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11 | nfcv 2616 |
. . . . 5
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12 | nfcsb1v 3414 |
. . . . 5
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13 | 11, 12 | nfxp 4977 |
. . . 4
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14 | sneq 3998 |
. . . . 5
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15 | csbeq1a 3407 |
. . . . 5
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16 | 14, 15 | xpeq12d 4976 |
. . . 4
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17 | 10, 13, 16 | cbviun 4318 |
. . 3
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18 | mpteq1 4483 |
. . 3
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19 | 17, 18 | ax-mp 5 |
. 2
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20 | nfcv 2616 |
. . 3
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21 | nfcv 2616 |
. . 3
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22 | nfcv 2616 |
. . 3
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23 | nfcsb1v 3414 |
. . 3
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24 | nfcv 2616 |
. . . 4
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25 | nfcsb1v 3414 |
. . . 4
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26 | 24, 25 | nfcsb 3416 |
. . 3
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27 | csbeq1a 3407 |
. . . 4
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28 | csbeq1a 3407 |
. . . 4
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29 | 27, 28 | sylan9eqr 2517 |
. . 3
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30 | 20, 12, 21, 22, 23, 26, 15, 29 | cbvmpt2x 6276 |
. 2
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31 | 9, 19, 30 | 3eqtr4ri 2494 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-iota 5492 df-fun 5531 df-fv 5537 df-oprab 6207 df-mpt2 6208 df-1st 6690 df-2nd 6691 |
This theorem is referenced by: mpt2mpts 6751 ovmptss 6767 |
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