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	$$⊢\mathrm{Rel}A\leftrightarrow A\circ \mathrm{I}=A$$

Proof

Step	Hyp	Ref	Expression
1		bj-dfrel3	$$⊢\mathrm{Rel}A\leftrightarrow {A\upharpoonright }_{\mathrm{V}}=A$$
2		coi1res	$$⊢A\circ \mathrm{I}={A\upharpoonright }_{\mathrm{V}}$$
3	2	eqeq1i	$$⊢A\circ \mathrm{I}=A\leftrightarrow {A\upharpoonright }_{\mathrm{V}}=A$$
4	1 3	bitr4i	$$⊢\mathrm{Rel}A\leftrightarrow A\circ \mathrm{I}=A$$