Theorem inxpss2

Description: Two ways to say that intersections with Cartesian products are in a subclass relation. (Contributed by Peter Mazsa, 8-Mar-2019)

	Ref	Expression
Assertion	inxpss2	⊢ ( ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ⊆ ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝑅 𝑦 → 𝑥 𝑆 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		relinxp	⊢ Rel ( 𝑅 ∩ ( 𝐴 × 𝐵 ) )
2		ssrel3	⊢ ( Rel ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) → ( ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ⊆ ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 → 𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ⊆ ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 → 𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ) )
4		inxpss3	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 → 𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝑅 𝑦 → 𝑥 𝑆 𝑦 ) )
5	3 4	bitri	⊢ ( ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ⊆ ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑥 𝑅 𝑦 → 𝑥 𝑆 𝑦 ) )