Theorem brelrn

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004)

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

Proof

Step	Hyp	Ref	Expression
1		brelrn.1	$$⊢A\in \mathrm{V}$$
2		brelrn.2	$$⊢B\in \mathrm{V}$$
3		brelrng	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\wedge ACB\right)\to B\in \mathrm{ran}C$$
4	1 2 3	mp3an12	$$⊢ACB\to B\in \mathrm{ran}C$$