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Metamath Proof Explorer

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 )