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Theorem mpomptsx 6088
 Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
mpomptsx
Distinct variable groups:   ,,,   ,,   ,
Allowed substitution hints:   ()   (,)

Proof of Theorem mpomptsx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2684 . . . . . 6
2 vex 2684 . . . . . 6
31, 2op1std 6039 . . . . 5
43csbeq1d 3005 . . . 4
51, 2op2ndd 6040 . . . . . 6
65csbeq1d 3005 . . . . 5
76csbeq2dv 3023 . . . 4
84, 7eqtrd 2170 . . 3
98mpomptx 5855 . 2
10 nfcv 2279 . . . 4
11 nfcv 2279 . . . . 5
12 nfcsb1v 3030 . . . . 5
1311, 12nfxp 4561 . . . 4
14 sneq 3533 . . . . 5
15 csbeq1a 3007 . . . . 5
1614, 15xpeq12d 4559 . . . 4
1710, 13, 16cbviun 3845 . . 3
18 mpteq1 4007 . . 3
1917, 18ax-mp 5 . 2
20 nfcv 2279 . . 3
21 nfcv 2279 . . 3
22 nfcv 2279 . . 3
23 nfcsb1v 3030 . . 3
24 nfcv 2279 . . . 4
25 nfcsb1v 3030 . . . 4
2624, 25nfcsb 3032 . . 3
27 csbeq1a 3007 . . . 4
28 csbeq1a 3007 . . . 4
2927, 28sylan9eqr 2192 . . 3
3020, 12, 21, 22, 23, 26, 15, 29cbvmpox 5842 . 2
319, 19, 303eqtr4ri 2169 1
 Colors of variables: wff set class Syntax hints:   wceq 1331  csb 2998  csn 3522  cop 3525  ciun 3808   cmpt 3984   cxp 4532  cfv 5118   cmpo 5769  c1st 6029  c2nd 6030 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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032 This theorem is referenced by:  mpompts  6089  mpofvex  6094
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