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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 ∈ 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