Theorem bj-elid5

Description: Characterization of the elements of _I . (Contributed by BJ, 22-Jun-2019)

	Ref	Expression
Assertion	bj-elid5	⊢ ( 𝐴 ∈ I ↔ ( 𝐴 ∈ ( V × V ) ∧ ( 1st ‘ 𝐴 ) = ( 2nd ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		reli	⊢ Rel I
2		df-rel	⊢ ( Rel I ↔ I ⊆ ( V × V ) )
3	1 2	mpbi	⊢ I ⊆ ( V × V )
4	3	sseli	⊢ ( 𝐴 ∈ I → 𝐴 ∈ ( V × V ) )
5		bj-elid4	⊢ ( 𝐴 ∈ ( V × V ) → ( 𝐴 ∈ I ↔ ( 1st ‘ 𝐴 ) = ( 2nd ‘ 𝐴 ) ) )
6	4 5	biadanii	⊢ ( 𝐴 ∈ I ↔ ( 𝐴 ∈ ( V × V ) ∧ ( 1st ‘ 𝐴 ) = ( 2nd ‘ 𝐴 ) ) )