Metamath Proof Explorer


Theorem inxpss3

Description: Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss ). (Contributed by Peter Mazsa, 8-Mar-2019)

Ref Expression
Assertion inxpss3 ( ∀ 𝑥𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ) ↔ ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) )

Proof

Step Hyp Ref Expression
1 brinxp2 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ↔ ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑅 𝑦 ) )
2 brinxp2 ( 𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ↔ ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑆 𝑦 ) )
3 1 2 imbi12i ( ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ) ↔ ( ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑅 𝑦 ) → ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑆 𝑦 ) ) )
4 imdistan ( ( ( 𝑥𝐴𝑦𝐵 ) → ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ) ↔ ( ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑅 𝑦 ) → ( ( 𝑥𝐴𝑦𝐵 ) ∧ 𝑥 𝑆 𝑦 ) ) )
5 3 4 bitr4i ( ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ) ↔ ( ( 𝑥𝐴𝑦𝐵 ) → ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ) )
6 5 2albii ( ∀ 𝑥𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ) ↔ ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦𝐵 ) → ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ) )
7 r2al ( ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ↔ ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦𝐵 ) → ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) ) )
8 6 7 bitr4i ( ∀ 𝑥𝑦 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐵 ) ) 𝑦𝑥 ( 𝑆 ∩ ( 𝐴 × 𝐵 ) ) 𝑦 ) ↔ ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝑅 𝑦𝑥 𝑆 𝑦 ) )