Theorem intop

Description: An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020)

	Ref	Expression
Assertion	intop	|- ( .o. e. ( M intOp N ) -> .o. : ( M X. M ) --> N )

Proof

Step	Hyp	Ref	Expression
1		df-intop	|- intOp = ( m e. _V , n e. _V |-> ( n ^m ( m X. m ) ) )
2	1	elmpocl	|- ( .o. e. ( M intOp N ) -> ( M e. _V /\ N e. _V ) )
3		intopval	|- ( ( M e. _V /\ N e. _V ) -> ( M intOp N ) = ( N ^m ( M X. M ) ) )
4	3	eleq2d	|- ( ( M e. _V /\ N e. _V ) -> ( .o. e. ( M intOp N ) <-> .o. e. ( N ^m ( M X. M ) ) ) )
5		elmapi	|- ( .o. e. ( N ^m ( M X. M ) ) -> .o. : ( M X. M ) --> N )
6	4 5	syl6bi	|- ( ( M e. _V /\ N e. _V ) -> ( .o. e. ( M intOp N ) -> .o. : ( M X. M ) --> N ) )
7	2 6	mpcom	|- ( .o. e. ( M intOp N ) -> .o. : ( M X. M ) --> N )