Metamath Proof Explorer


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 eqtrid
 |-  ( A e. dom vol -> ( vol ` A ) = ( vol* ` A ) )