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	⊢ ( ( 𝐴 ∩ 𝐶 ) = ∅ → ( ( 𝐴 × 𝐵 ) ↾ 𝐶 ) = ∅ )

Proof

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