Theorem bj-dfrel3

Description: Alternate definition of relation: a class is a relation if and only if it is equal to its "relation part" (see relpart ). (Contributed by NM, 14-May-2008) Do not depend on df-cnv to move it earlier. (Revised by BJ, 16-Oct-2026)

	Ref	Expression
Assertion	bj-dfrel3	$$⊢\mathrm{Rel}R\leftrightarrow {R\upharpoonright }_{\mathrm{V}}=R$$

Proof

Step	Hyp	Ref	Expression
1		dfss2	$$⊢R\subseteq \mathrm{V}\times \mathrm{V}\leftrightarrow R\cap \left(\mathrm{V}\times \mathrm{V}\right)=R$$
2		df-rel	$$⊢\mathrm{Rel}R\leftrightarrow R\subseteq \mathrm{V}\times \mathrm{V}$$
3		df-res	$$⊢{R\upharpoonright }_{\mathrm{V}}=R\cap \left(\mathrm{V}\times \mathrm{V}\right)$$
4	3	eqeq1i	$$⊢{R\upharpoonright }_{\mathrm{V}}=R\leftrightarrow R\cap \left(\mathrm{V}\times \mathrm{V}\right)=R$$
5	1 2 4	3bitr4i	$$⊢\mathrm{Rel}R\leftrightarrow {R\upharpoonright }_{\mathrm{V}}=R$$