Theorem bj-coi1

Description: The identity relation is a right-unit for composition. Part of Theorem 3.7(i) of Monk1 p. 36. (Contributed by NM, 22-Apr-2004) Strengthen to biconditional form and shorten (Revised by BJ, 16-Oct-2026)

	Ref	Expression
Assertion	bj-coi1	⊢ ( Rel 𝐴 ↔ ( 𝐴 ∘ I ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		bj-dfrel3	⊢ ( Rel 𝐴 ↔ ( 𝐴 ↾ V ) = 𝐴 )
2		coi1res	⊢ ( 𝐴 ∘ I ) = ( 𝐴 ↾ V )
3	2	eqeq1i	⊢ ( ( 𝐴 ∘ I ) = 𝐴 ↔ ( 𝐴 ↾ V ) = 𝐴 )
4	1 3	bitr4i	⊢ ( Rel 𝐴 ↔ ( 𝐴 ∘ I ) = 𝐴 )