Metamath Proof Explorer


Theorem inxpss

Description: Two ways to say that an intersection with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019)

Ref Expression
Assertion inxpss ( ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ⊆ 𝑆 ↔ ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) )

Proof

Step Hyp Ref Expression
1 brinxp2 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ↔ ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑅 𝑦 ) )
2 1 imbi1i ( ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 𝑆 𝑦 ) ↔ ( ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑅 𝑦 ) → 𝑥 𝑆 𝑦 ) )
3 impexp ( ( ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑅 𝑦 ) → 𝑥 𝑆 𝑦 ) ↔ ( ( 𝑥𝐴𝑦𝐵 ) → ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ) )
4 2 3 bitri ( ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 𝑆 𝑦 ) ↔ ( ( 𝑥𝐴𝑦𝐵 ) → ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ) )
5 4 2albii ( ∀ 𝑥𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 𝑆 𝑦 ) ↔ ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦𝐵 ) → ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ) )
6 relinxp Rel ( 𝑅 ∩ ( 𝐴 × 𝐵 ) )
7 ssrel3 ( Rel ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) → ( ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ⊆ 𝑆 ↔ ∀ 𝑥𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 𝑆 𝑦 ) ) )
8 6 7 ax-mp ( ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ⊆ 𝑆 ↔ ∀ 𝑥𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 𝑆 𝑦 ) )
9 r2al ( ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ↔ ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦𝐵 ) → ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ) )
10 5 8 9 3bitr4i ( ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) ⊆ 𝑆 ↔ ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) )