Theorem mptfvmpt

Description: A function in maps-to notation as the value of another function in maps-to notation. (Contributed by AV, 20-Aug-2022)

	Ref	Expression
Hypotheses	mptfvmpt.y	|- ( y = Y -> M = ( x e. V |-> A ) )
	mptfvmpt.g	|- G = ( y e. W |-> M )
	mptfvmpt.v	|- V = ( F ` X )
Assertion	mptfvmpt	|- ( Y e. W -> ( G ` Y ) = ( x e. V |-> A ) )

Proof

Step	Hyp	Ref	Expression
1		mptfvmpt.y	|- ( y = Y -> M = ( x e. V |-> A ) )
2		mptfvmpt.g	|- G = ( y e. W |-> M )
3		mptfvmpt.v	|- V = ( F ` X )
4	3	fvexi	|- V e. _V
5	4	mptex	|- ( x e. V |-> A ) e. _V
6	1 2 5	fvmpt	|- ( Y e. W -> ( G ` Y ) = ( x e. V |-> A ) )