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Mirrors > Home > ILE Home > Th. List > mpomptsx | 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 |
---|---|
mpomptsx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . . . . 6 | |
2 | vex 2689 | . . . . . 6 | |
3 | 1, 2 | op1std 6046 | . . . . 5 |
4 | 3 | csbeq1d 3010 | . . . 4 |
5 | 1, 2 | op2ndd 6047 | . . . . . 6 |
6 | 5 | csbeq1d 3010 | . . . . 5 |
7 | 6 | csbeq2dv 3028 | . . . 4 |
8 | 4, 7 | eqtrd 2172 | . . 3 |
9 | 8 | mpomptx 5862 | . 2 |
10 | nfcv 2281 | . . . 4 | |
11 | nfcv 2281 | . . . . 5 | |
12 | nfcsb1v 3035 | . . . . 5 | |
13 | 11, 12 | nfxp 4566 | . . . 4 |
14 | sneq 3538 | . . . . 5 | |
15 | csbeq1a 3012 | . . . . 5 | |
16 | 14, 15 | xpeq12d 4564 | . . . 4 |
17 | 10, 13, 16 | cbviun 3850 | . . 3 |
18 | mpteq1 4012 | . . 3 | |
19 | 17, 18 | ax-mp 5 | . 2 |
20 | nfcv 2281 | . . 3 | |
21 | nfcv 2281 | . . 3 | |
22 | nfcv 2281 | . . 3 | |
23 | nfcsb1v 3035 | . . 3 | |
24 | nfcv 2281 | . . . 4 | |
25 | nfcsb1v 3035 | . . . 4 | |
26 | 24, 25 | nfcsb 3037 | . . 3 |
27 | csbeq1a 3012 | . . . 4 | |
28 | csbeq1a 3012 | . . . 4 | |
29 | 27, 28 | sylan9eqr 2194 | . . 3 |
30 | 20, 12, 21, 22, 23, 26, 15, 29 | cbvmpox 5849 | . 2 |
31 | 9, 19, 30 | 3eqtr4ri 2171 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 csb 3003 csn 3527 cop 3530 ciun 3813 cmpt 3989 cxp 4537 cfv 5123 cmpo 5776 c1st 6036 c2nd 6037 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: mpompts 6096 mpofvex 6101 |
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