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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  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
Distinct variable groups:    x, y, z, A    y, B, z   
z, C
Allowed substitution hints:    B( x)    C( x, y)

Proof of Theorem mpomptsx
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2684 . . . . . 6  |-  u  e. 
_V
2 vex 2684 . . . . . 6  |-  v  e. 
_V
31, 2op1std 6039 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  ( 1st `  z
)  =  u )
43csbeq1d 3005 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
51, 2op2ndd 6040 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( 2nd `  z
)  =  v )
65csbeq1d 3005 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 2nd `  z
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
76csbeq2dv 3023 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
84, 7eqtrd 2170 . . 3  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
98mpomptx 5855 . 2  |-  ( z  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C )  =  ( u  e.  A ,  v  e.  [_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
10 nfcv 2279 . . . 4  |-  F/_ u
( { x }  X.  B )
11 nfcv 2279 . . . . 5  |-  F/_ x { u }
12 nfcsb1v 3030 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
1311, 12nfxp 4561 . . . 4  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
14 sneq 3533 . . . . 5  |-  ( x  =  u  ->  { x }  =  { u } )
15 csbeq1a 3007 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
1614, 15xpeq12d 4559 . . . 4  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
1710, 13, 16cbviun 3845 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
18 mpteq1 4007 . . 3  |-  ( U_ x  e.  A  ( { x }  X.  B )  =  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  ->  (
z  e.  U_ x  e.  A  ( {
x }  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )  =  ( z  e.  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C ) )
1917, 18ax-mp 5 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C )  =  ( z  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
20 nfcv 2279 . . 3  |-  F/_ u B
21 nfcv 2279 . . 3  |-  F/_ u C
22 nfcv 2279 . . 3  |-  F/_ v C
23 nfcsb1v 3030 . . 3  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
24 nfcv 2279 . . . 4  |-  F/_ y
u
25 nfcsb1v 3030 . . . 4  |-  F/_ y [_ v  /  y ]_ C
2624, 25nfcsb 3032 . . 3  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
27 csbeq1a 3007 . . . 4  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
28 csbeq1a 3007 . . . 4  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
2927, 28sylan9eqr 2192 . . 3  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
3020, 12, 21, 22, 23, 26, 15, 29cbvmpox 5842 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( u  e.  A ,  v  e. 
[_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
319, 19, 303eqtr4ri 2169 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
Colors of variables: wff set class
Syntax hints:    = wceq 1331   [_csb 2998   {csn 3522   <.cop 3525   U_ciun 3808    |-> cmpt 3984    X. cxp 4532   ` cfv 5118    e. cmpo 5769   1stc1st 6029   2ndc2nd 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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