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Metamath Proof Explorer

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 ) )