Metamath Proof Explorer


Theorem ofmresex

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

Ref Expression
Hypotheses ofmresex.a ( 𝜑𝐴𝑉 )
ofmresex.b ( 𝜑𝐵𝑊 )
Assertion ofmresex ( 𝜑 → ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) ∈ V )

Proof

Step Hyp Ref Expression
1 ofmresex.a ( 𝜑𝐴𝑉 )
2 ofmresex.b ( 𝜑𝐵𝑊 )
3 1 2 xpexd ( 𝜑 → ( 𝐴 × 𝐵 ) ∈ V )
4 ofexg ( ( 𝐴 × 𝐵 ) ∈ V → ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) ∈ V )
5 3 4 syl ( 𝜑 → ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) ∈ V )