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 ∈ 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 ∈ dom vol → ( ( vol* ↾ dom vol ) ‘ 𝐴 ) = ( vol* ‘ 𝐴 ) )
4	2 3	eqtrid	⊢ ( 𝐴 ∈ dom vol → ( vol ‘ 𝐴 ) = ( vol* ‘ 𝐴 ) )