Theorem fvresd

Description: The value of a restricted function, deduction version of fvres . (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypothesis	fvresd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
Assertion	fvresd	⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fvresd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		fvres	⊢ ( 𝐴 ∈ 𝐵 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
3	1 2	syl	⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )