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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . . . 6 | |
2 | vex 2560 | . . . . . 6 | |
3 | 1, 2 | op1std 5775 | . . . . 5 |
4 | 3 | csbeq1d 2858 | . . . 4 |
5 | 1, 2 | op2ndd 5776 | . . . . . 6 |
6 | 5 | csbeq1d 2858 | . . . . 5 |
7 | 6 | csbeq2dv 2875 | . . . 4 |
8 | 4, 7 | eqtrd 2072 | . . 3 |
9 | 8 | mpt2mptx 5595 | . 2 |
10 | nfcv 2178 | . . . 4 | |
11 | nfcv 2178 | . . . . 5 | |
12 | nfcsb1v 2882 | . . . . 5 | |
13 | 11, 12 | nfxp 4371 | . . . 4 |
14 | sneq 3386 | . . . . 5 | |
15 | csbeq1a 2860 | . . . . 5 | |
16 | 14, 15 | xpeq12d 4370 | . . . 4 |
17 | 10, 13, 16 | cbviun 3694 | . . 3 |
18 | mpteq1 3841 | . . 3 | |
19 | 17, 18 | ax-mp 7 | . 2 |
20 | nfcv 2178 | . . 3 | |
21 | nfcv 2178 | . . 3 | |
22 | nfcv 2178 | . . 3 | |
23 | nfcsb1v 2882 | . . 3 | |
24 | nfcv 2178 | . . . 4 | |
25 | nfcsb1v 2882 | . . . 4 | |
26 | 24, 25 | nfcsb 2884 | . . 3 |
27 | csbeq1a 2860 | . . . 4 | |
28 | csbeq1a 2860 | . . . 4 | |
29 | 27, 28 | sylan9eqr 2094 | . . 3 |
30 | 20, 12, 21, 22, 23, 26, 15, 29 | cbvmpt2x 5582 | . 2 |
31 | 9, 19, 30 | 3eqtr4ri 2071 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1243 csb 2852 csn 3375 cop 3378 ciun 3657 cmpt 3818 cxp 4343 cfv 4902 cmpt2 5514 c1st 5765 c2nd 5766 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fv 4910 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: mpt2mpts 5824 mpt2fvex 5829 |
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