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Metamath Proof Explorer

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 ) )