Theorem mblvol

Description: The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014)

	Ref	Expression
Assertion	mblvol	|- ( A e. dom vol -> ( vol ` A ) = ( vol* ` A ) )

Proof

Step	Hyp	Ref	Expression
1		volres	|- vol = ( vol* |` dom vol )
2	1	fveq1i	|- ( vol ` A ) = ( ( vol* |` dom vol ) ` A )
3		fvres	|- ( A e. dom vol -> ( ( vol* |` dom vol ) ` A ) = ( vol* ` A ) )
4	2 3	syl5eq	|- ( A e. dom vol -> ( vol ` A ) = ( vol* ` A ) )