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Theorem mpt2mptsx 5765
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  ,  |->  C 
U_  { }  X.  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C
Distinct variable groups:   ,,,   ,,   , C
Allowed substitution hints:   ()    C(,)

Proof of Theorem mpt2mptsx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . 6 
_V
2 vex 2554 . . . . . 6 
_V
31, 2op1std 5717 . . . . 5  <. , 
>.  1st `
43csbeq1d 2852 . . . 4  <. , 
>.  [_ 1st `  ]_
[_ 2nd `  ]_ C  [_  ]_
[_ 2nd `  ]_ C
51, 2op2ndd 5718 . . . . . 6  <. , 
>.  2nd `
65csbeq1d 2852 . . . . 5  <. , 
>.  [_ 2nd `  ]_ C  [_  ]_ C
76csbeq2dv 2869 . . . 4  <. , 
>.  [_  ]_ [_ 2nd `  ]_ C  [_  ]_ [_  ]_ C
84, 7eqtrd 2069 . . 3  <. , 
>.  [_ 1st `  ]_
[_ 2nd `  ]_ C  [_  ]_
[_  ]_ C
98mpt2mptx 5537 . 2  U_  { }  X.  [_  ]_  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C  ,  [_  ]_  |->  [_  ]_ [_  ]_ C
10 nfcv 2175 . . . 4  F/_ { }  X.
11 nfcv 2175 . . . . 5  F/_ { }
12 nfcsb1v 2876 . . . . 5  F/_ [_  ]_
1311, 12nfxp 4314 . . . 4  F/_ { }  X.  [_  ]_
14 sneq 3378 . . . . 5  { }  { }
15 csbeq1a 2854 . . . . 5  [_  ]_
1614, 15xpeq12d 4313 . . . 4  { }  X.  { }  X.  [_  ]_
1710, 13, 16cbviun 3685 . . 3  U_  { }  X. 
U_  { }  X.  [_  ]_
18 mpteq1 3832 . . 3  U_  { }  X.  U_  { }  X.  [_  ]_  U_  { }  X.  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C  U_  { }  X.  [_  ]_  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C
1917, 18ax-mp 7 . 2  U_  { }  X.  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C  U_  { }  X.  [_  ]_  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C
20 nfcv 2175 . . 3  F/_
21 nfcv 2175 . . 3  F/_ C
22 nfcv 2175 . . 3  F/_ C
23 nfcsb1v 2876 . . 3  F/_ [_  ]_ [_  ]_ C
24 nfcv 2175 . . . 4  F/_
25 nfcsb1v 2876 . . . 4  F/_ [_  ]_ C
2624, 25nfcsb 2878 . . 3  F/_ [_  ]_
[_  ]_ C
27 csbeq1a 2854 . . . 4  C  [_  ]_ C
28 csbeq1a 2854 . . . 4  [_  ]_ C 
[_  ]_
[_  ]_ C
2927, 28sylan9eqr 2091 . . 3  C  [_  ]_ [_  ]_ C
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 5524 . 2  ,  |->  C  , 
[_  ]_  |->  [_  ]_
[_  ]_ C
319, 19, 303eqtr4ri 2068 1  ,  |->  C 
U_  { }  X.  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C
Colors of variables: wff set class
Syntax hints:   wceq 1242   [_csb 2846   {csn 3367   <.cop 3370   U_ciun 3648    |-> cmpt 3809    X. cxp 4286   ` cfv 4845    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  mpt2mpts  5766  mpt2fvex  5771
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