Theorem inxpex

Description: Sufficient condition for an intersection with a Cartesian product to be a set. (Contributed by Peter Mazsa, 10-May-2019)

	Ref	Expression
Assertion	inxpex	⊢ ( ( 𝑅 ∈ 𝑊 ∨ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) → ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		inex1g	⊢ ( 𝑅 ∈ 𝑊 → ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ∈ V )
2		xpexg	⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 × 𝐵 ) ∈ V )
3		inex2g	⊢ ( ( 𝐴 × 𝐵 ) ∈ V → ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ∈ V )
4	2 3	syl	⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) → ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ∈ V )
5	1 4	jaoi	⊢ ( ( 𝑅 ∈ 𝑊 ∨ ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ) → ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ∈ V )