Theorem stafval

Description: The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015)

	Ref	Expression
Hypotheses	staffval.b	|- B = ( Base ` R )
	staffval.i	|- .* = ( *r ` R )
	staffval.f	|- .xb = ( *rf ` R )
Assertion	stafval	|- ( A e. B -> ( .xb ` A ) = ( .* ` A ) )

Proof

Step	Hyp	Ref	Expression
1		staffval.b	|- B = ( Base ` R )
2		staffval.i	|- .* = ( *r ` R )
3		staffval.f	|- .xb = ( *rf ` R )
4		fveq2	|- ( x = A -> ( .* ` x ) = ( .* ` A ) )
5	1 2 3	staffval	|- .xb = ( x e. B |-> ( .* ` x ) )
6		fvex	|- ( .* ` A ) e. _V
7	4 5 6	fvmpt	|- ( A e. B -> ( .xb ` A ) = ( .* ` A ) )