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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 ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
ofmresex.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion ofmresex ( 𝜑 → ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) ∈ V )

Proof

Step Hyp Ref Expression
1 ofmresex.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2 ofmresex.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
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 )