Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpt2mptsx Unicode version

Theorem mpt2mptsx 5823
 Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
mpt2mptsx
Distinct variable groups:   ,,,   ,,   ,
Allowed substitution hints:   ()   (,)

Proof of Theorem mpt2mptsx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . . 6
2 vex 2560 . . . . . 6
31, 2op1std 5775 . . . . 5
43csbeq1d 2858 . . . 4
51, 2op2ndd 5776 . . . . . 6
65csbeq1d 2858 . . . . 5
76csbeq2dv 2875 . . . 4
84, 7eqtrd 2072 . . 3
98mpt2mptx 5595 . 2
10 nfcv 2178 . . . 4
11 nfcv 2178 . . . . 5
12 nfcsb1v 2882 . . . . 5
1311, 12nfxp 4371 . . . 4
14 sneq 3386 . . . . 5
15 csbeq1a 2860 . . . . 5
1614, 15xpeq12d 4370 . . . 4
1710, 13, 16cbviun 3694 . . 3
18 mpteq1 3841 . . 3
1917, 18ax-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
2624, 25nfcsb 2884 . . 3
27 csbeq1a 2860 . . . 4
28 csbeq1a 2860 . . . 4
2927, 28sylan9eqr 2094 . . 3
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 5582 . 2
319, 19, 303eqtr4ri 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
 Copyright terms: Public domain W3C validator