Theorem opelrn

Description: Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997)

	Ref	Expression
Hypotheses	brelrn.1	$$⊢A\in \mathrm{V}$$
	brelrn.2	$$⊢B\in \mathrm{V}$$
Assertion	opelrn	$$⊢⟨A,B⟩\in C\to B\in \mathrm{ran}C$$

Proof

Step	Hyp	Ref	Expression
1		brelrn.1	$$⊢A\in \mathrm{V}$$
2		brelrn.2	$$⊢B\in \mathrm{V}$$
3		df-br	$$⊢ACB\leftrightarrow ⟨A,B⟩\in C$$
4	1 2	brelrn	$$⊢ACB\to B\in \mathrm{ran}C$$
5	3 4	sylbir	$$⊢⟨A,B⟩\in C\to B\in \mathrm{ran}C$$