Theorem fssresd

Description: Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	fssresd.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
	fssresd.2	⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
Assertion	fssresd	⊢ ( 𝜑 → ( 𝐹 ↾ 𝐶 ) : 𝐶 ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		fssresd.1	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
2		fssresd.2	⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
3		fssres	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 ⟶ 𝐵 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐹 ↾ 𝐶 ) : 𝐶 ⟶ 𝐵 )