Theorem ssres

Description: Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994)

	Ref	Expression
Assertion	ssres	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ↾ 𝐶 ) ⊆ ( 𝐵 ↾ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssrin	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ ( 𝐶 × V ) ) ⊆ ( 𝐵 ∩ ( 𝐶 × V ) ) )
2		df-res	⊢ ( 𝐴 ↾ 𝐶 ) = ( 𝐴 ∩ ( 𝐶 × V ) )
3		df-res	⊢ ( 𝐵 ↾ 𝐶 ) = ( 𝐵 ∩ ( 𝐶 × V ) )
4	1 2 3	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ↾ 𝐶 ) ⊆ ( 𝐵 ↾ 𝐶 ) )