Metamath Proof Explorer


Theorem idinxpres

Description: The intersection of the identity relation with a cartesian product is the restriction of the identity relation to the intersection of the factors. (Contributed by FL, 2-Aug-2009) (Proof shortened by Peter Mazsa, 9-Sep-2022) Generalize statement from cartesian square (now idinxpresid ) to cartesian product. (Revised by BJ, 23-Dec-2023)

Ref Expression
Assertion idinxpres ( I ∩ ( 𝐴 × 𝐵 ) ) = ( I ↾ ( 𝐴𝐵 ) )

Proof

Step Hyp Ref Expression
1 elidinxp ( 𝑥 ∈ ( I ∩ ( 𝐴 × 𝐵 ) ) ↔ ∃ 𝑦 ∈ ( 𝐴𝐵 ) 𝑥 = ⟨ 𝑦 , 𝑦 ⟩ )
2 elrid ( 𝑥 ∈ ( I ↾ ( 𝐴𝐵 ) ) ↔ ∃ 𝑦 ∈ ( 𝐴𝐵 ) 𝑥 = ⟨ 𝑦 , 𝑦 ⟩ )
3 1 2 bitr4i ( 𝑥 ∈ ( I ∩ ( 𝐴 × 𝐵 ) ) ↔ 𝑥 ∈ ( I ↾ ( 𝐴𝐵 ) ) )
4 3 eqriv ( I ∩ ( 𝐴 × 𝐵 ) ) = ( I ↾ ( 𝐴𝐵 ) )