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Theorem mpt2mpts 6848
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
Assertion
Ref Expression
mpt2mpts  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
Distinct variable groups:    x, y,
z, A    y, B, z    z, C    x, B
Allowed substitution hints:    C( x, y)

Proof of Theorem mpt2mpts
StepHypRef Expression
1 mpt2mptsx 6847 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2 iunxpconst 5056 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
3 mpteq1 4527 . . 3  |-  ( U_ x  e.  A  ( { x }  X.  B )  =  ( A  X.  B )  ->  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )  =  ( z  e.  ( A  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C ) )
42, 3ax-mp 5 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C )  =  ( z  e.  ( A  X.  B ) 
|->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
51, 4eqtri 2496 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   [_csb 3435   {csn 4027   U_ciun 4325    |-> cmpt 4505    X. cxp 4997   ` cfv 5588    |-> cmpt2 6286   1stc1st 6782   2ndc2nd 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785
This theorem is referenced by:  offval22  6862  dfmpt2  6873  mpt2sn  6874  matgsum  18734
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