Metamath Proof Explorer


Theorem bj-elid7

Description: Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019)

Ref Expression
Assertion bj-elid7 ( ⟨ 𝐵 , 𝐶 ⟩ ∈ ( I ↾ 𝐴 ) ↔ ( 𝐵𝐴𝐵 = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 df-br ( 𝐵 ( I ↾ 𝐴 ) 𝐶 ↔ ⟨ 𝐵 , 𝐶 ⟩ ∈ ( I ↾ 𝐴 ) )
2 bj-idreseqb ( 𝐵 ( I ↾ 𝐴 ) 𝐶 ↔ ( 𝐵𝐴𝐵 = 𝐶 ) )
3 1 2 bitr3i ( ⟨ 𝐵 , 𝐶 ⟩ ∈ ( I ↾ 𝐴 ) ↔ ( 𝐵𝐴𝐵 = 𝐶 ) )