Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2mptsx Structured version   Unicode version

Theorem mpt2mptsx 6844
 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 3096 . . . . . 6
2 vex 3096 . . . . . 6
31, 2op1std 6791 . . . . 5
43csbeq1d 3424 . . . 4
51, 2op2ndd 6792 . . . . . 6
65csbeq1d 3424 . . . . 5
76csbeq2dv 3817 . . . 4
84, 7eqtrd 2482 . . 3
98mpt2mptx 6374 . 2
10 nfcv 2603 . . . 4
11 nfcv 2603 . . . . 5
12 nfcsb1v 3433 . . . . 5
1311, 12nfxp 5012 . . . 4
14 sneq 4020 . . . . 5
15 csbeq1a 3426 . . . . 5
1614, 15xpeq12d 5010 . . . 4
1710, 13, 16cbviun 4348 . . 3
18 mpteq1 4513 . . 3
1917, 18ax-mp 5 . 2
20 nfcv 2603 . . 3
21 nfcv 2603 . . 3
22 nfcv 2603 . . 3
23 nfcsb1v 3433 . . 3
24 nfcv 2603 . . . 4
25 nfcsb1v 3433 . . . 4
2624, 25nfcsb 3435 . . 3
27 csbeq1a 3426 . . . 4
28 csbeq1a 3426 . . . 4
2927, 28sylan9eqr 2504 . . 3
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 6356 . 2
319, 19, 303eqtr4ri 2481 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1381  csb 3417  csn 4010  cop 4016  ciun 4311   cmpt 4491   cxp 4983  cfv 5574   cmpt2 6279  c1st 6779  c2nd 6780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fv 5582  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782 This theorem is referenced by:  mpt2mpts  6845  ovmptss  6862
 Copyright terms: Public domain W3C validator