Theorem bj-eldiag

Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 . (Contributed by BJ, 22-Jun-2019)

	Ref	Expression
Assertion	bj-eldiag	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( Id ‘ 𝐴 ) ↔ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ ( 1st ‘ 𝐵 ) = ( 2nd ‘ 𝐵 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-diagval2	⊢ ( 𝐴 ∈ 𝑉 → ( Id ‘ 𝐴 ) = ( I ∩ ( 𝐴 × 𝐴 ) ) )
2	1	eleq2d	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( Id ‘ 𝐴 ) ↔ 𝐵 ∈ ( I ∩ ( 𝐴 × 𝐴 ) ) ) )
3		elin	⊢ ( 𝐵 ∈ ( I ∩ ( 𝐴 × 𝐴 ) ) ↔ ( 𝐵 ∈ I ∧ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) )
4		ancom	⊢ ( ( 𝐵 ∈ I ∧ 𝐵 ∈ ( 𝐴 × 𝐴 ) ) ↔ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ 𝐵 ∈ I ) )
5		bj-elid4	⊢ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) → ( 𝐵 ∈ I ↔ ( 1st ‘ 𝐵 ) = ( 2nd ‘ 𝐵 ) ) )
6	5	pm5.32i	⊢ ( ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ 𝐵 ∈ I ) ↔ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ ( 1st ‘ 𝐵 ) = ( 2nd ‘ 𝐵 ) ) )
7	3 4 6	3bitri	⊢ ( 𝐵 ∈ ( I ∩ ( 𝐴 × 𝐴 ) ) ↔ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ ( 1st ‘ 𝐵 ) = ( 2nd ‘ 𝐵 ) ) )
8	2 7	bitrdi	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( Id ‘ 𝐴 ) ↔ ( 𝐵 ∈ ( 𝐴 × 𝐴 ) ∧ ( 1st ‘ 𝐵 ) = ( 2nd ‘ 𝐵 ) ) ) )