Theorem volres

Description: A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014)

	Ref	Expression
Assertion	volres	⊢ vol = ( vol* ↾ dom vol )

Proof

Step	Hyp	Ref	Expression
1		resdmres	⊢ ( vol* ↾ dom ( vol* ↾ { 𝑥 ∣ ∀ 𝑦 ∈ ( ◡ vol* “ ℝ ) ( vol* ‘ 𝑦 ) = ( ( vol* ‘ ( 𝑦 ∩ 𝑥 ) ) + ( vol* ‘ ( 𝑦 ∖ 𝑥 ) ) ) } ) ) = ( vol* ↾ { 𝑥 ∣ ∀ 𝑦 ∈ ( ◡ vol* “ ℝ ) ( vol* ‘ 𝑦 ) = ( ( vol* ‘ ( 𝑦 ∩ 𝑥 ) ) + ( vol* ‘ ( 𝑦 ∖ 𝑥 ) ) ) } )
2		df-vol	⊢ vol = ( vol* ↾ { 𝑥 ∣ ∀ 𝑦 ∈ ( ◡ vol* “ ℝ ) ( vol* ‘ 𝑦 ) = ( ( vol* ‘ ( 𝑦 ∩ 𝑥 ) ) + ( vol* ‘ ( 𝑦 ∖ 𝑥 ) ) ) } )
3	2	dmeqi	⊢ dom vol = dom ( vol* ↾ { 𝑥 ∣ ∀ 𝑦 ∈ ( ◡ vol* “ ℝ ) ( vol* ‘ 𝑦 ) = ( ( vol* ‘ ( 𝑦 ∩ 𝑥 ) ) + ( vol* ‘ ( 𝑦 ∖ 𝑥 ) ) ) } )
4	3	reseq2i	⊢ ( vol* ↾ dom vol ) = ( vol* ↾ dom ( vol* ↾ { 𝑥 ∣ ∀ 𝑦 ∈ ( ◡ vol* “ ℝ ) ( vol* ‘ 𝑦 ) = ( ( vol* ‘ ( 𝑦 ∩ 𝑥 ) ) + ( vol* ‘ ( 𝑦 ∖ 𝑥 ) ) ) } ) )
5	1 4 2	3eqtr4ri	⊢ vol = ( vol* ↾ dom vol )