Theorem resres

Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008)

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

Proof

Step	Hyp	Ref	Expression
1		df-res	⊢ ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ↾ 𝐵 ) ∩ ( 𝐶 × V ) )
2		df-res	⊢ ( 𝐴 ↾ 𝐵 ) = ( 𝐴 ∩ ( 𝐵 × V ) )
3	2	ineq1i	⊢ ( ( 𝐴 ↾ 𝐵 ) ∩ ( 𝐶 × V ) ) = ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∩ ( 𝐶 × V ) )
4		xpindir	⊢ ( ( 𝐵 ∩ 𝐶 ) × V ) = ( ( 𝐵 × V ) ∩ ( 𝐶 × V ) )
5	4	ineq2i	⊢ ( 𝐴 ∩ ( ( 𝐵 ∩ 𝐶 ) × V ) ) = ( 𝐴 ∩ ( ( 𝐵 × V ) ∩ ( 𝐶 × V ) ) )
6		df-res	⊢ ( 𝐴 ↾ ( 𝐵 ∩ 𝐶 ) ) = ( 𝐴 ∩ ( ( 𝐵 ∩ 𝐶 ) × V ) )
7		inass	⊢ ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∩ ( 𝐶 × V ) ) = ( 𝐴 ∩ ( ( 𝐵 × V ) ∩ ( 𝐶 × V ) ) )
8	5 6 7	3eqtr4ri	⊢ ( ( 𝐴 ∩ ( 𝐵 × V ) ) ∩ ( 𝐶 × V ) ) = ( 𝐴 ↾ ( 𝐵 ∩ 𝐶 ) )
9	1 3 8	3eqtri	⊢ ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ ( 𝐵 ∩ 𝐶 ) )