Theorem inres

Description: Move intersection into class restriction. (Contributed by NM, 18-Dec-2008)

	Ref	Expression
Assertion	inres	⊢ ( 𝐴 ∩ ( 𝐵 ↾ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		inass	⊢ ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐶 × V ) ) = ( 𝐴 ∩ ( 𝐵 ∩ ( 𝐶 × V ) ) )
2		df-res	⊢ ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ∩ 𝐵 ) ∩ ( 𝐶 × V ) )
3		df-res	⊢ ( 𝐵 ↾ 𝐶 ) = ( 𝐵 ∩ ( 𝐶 × V ) )
4	3	ineq2i	⊢ ( 𝐴 ∩ ( 𝐵 ↾ 𝐶 ) ) = ( 𝐴 ∩ ( 𝐵 ∩ ( 𝐶 × V ) ) )
5	1 2 4	3eqtr4ri	⊢ ( 𝐴 ∩ ( 𝐵 ↾ 𝐶 ) ) = ( ( 𝐴 ∩ 𝐵 ) ↾ 𝐶 )