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Metamath Proof Explorer

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 ) )