Theorem ofmresval

Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014)

	Ref	Expression
Hypotheses	ofmresval.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐴 )
	ofmresval.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
Assertion	ofmresval	⊢ ( 𝜑 → ( 𝐹 ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) 𝐺 ) = ( 𝐹 ∘f 𝑅 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		ofmresval.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐴 )
2		ofmresval.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
3		ovres	⊢ ( ( 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) 𝐺 ) = ( 𝐹 ∘f 𝑅 𝐺 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐹 ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) 𝐺 ) = ( 𝐹 ∘f 𝑅 𝐺 ) )