Theorem elmapg

Description: Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006) (Revised by Mario Carneiro, 15-Nov-2014)

	Ref	Expression
Assertion	elmapg	|- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^m B ) <-> C : B --> A ) )

Proof

Step	Hyp	Ref	Expression
1		mapvalg	|- ( ( A e. V /\ B e. W ) -> ( A ^m B ) = { g | g : B --> A } )
2	1	eleq2d	|- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^m B ) <-> C e. { g | g : B --> A } ) )
3		fex2	|- ( ( C : B --> A /\ B e. W /\ A e. V ) -> C e. _V )
4	3	3com13	|- ( ( A e. V /\ B e. W /\ C : B --> A ) -> C e. _V )
5	4	3expia	|- ( ( A e. V /\ B e. W ) -> ( C : B --> A -> C e. _V ) )
6		feq1	|- ( g = C -> ( g : B --> A <-> C : B --> A ) )
7	6	elab3g	|- ( ( C : B --> A -> C e. _V ) -> ( C e. { g | g : B --> A } <-> C : B --> A ) )
8	5 7	syl	|- ( ( A e. V /\ B e. W ) -> ( C e. { g | g : B --> A } <-> C : B --> A ) )
9	2 8	bitrd	|- ( ( A e. V /\ B e. W ) -> ( C e. ( A ^m B ) <-> C : B --> A ) )