Metamath Proof Explorer


Theorem xpdisjres

Description: Restriction of a constant function (or other Cartesian product) outside of its domain. (Contributed by Thierry Arnoux, 25-Jan-2017)

Ref Expression
Assertion xpdisjres ( ( 𝐴𝐶 ) = ∅ → ( ( 𝐴 × 𝐵 ) ↾ 𝐶 ) = ∅ )

Proof

Step Hyp Ref Expression
1 df-res ( ( 𝐴 × 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 × 𝐵 ) ∩ ( 𝐶 × V ) )
2 xpdisj1 ( ( 𝐴𝐶 ) = ∅ → ( ( 𝐴 × 𝐵 ) ∩ ( 𝐶 × V ) ) = ∅ )
3 1 2 syl5eq ( ( 𝐴𝐶 ) = ∅ → ( ( 𝐴 × 𝐵 ) ↾ 𝐶 ) = ∅ )