Theorem mpompts

Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015)

	Ref	Expression
Assertion	mpompts	|- ( x e. A , y e. B |-> C ) = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )

Proof

Step	Hyp	Ref	Expression
1		mpomptsx	|- ( 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	|- U_ x e. A ( { x } X. B ) = ( A X. B )
3	2	mpteq1i	|- ( 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 )
4	1 3	eqtri	|- ( x e. A , y e. B |-> C ) = ( z e. ( A X. B ) |-> [_ ( 1st ` z ) / x ]_ [_ ( 2nd ` z ) / y ]_ C )