Theorem relelrn

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008)

	Ref	Expression
Assertion	relelrn	⊢ ( ( Rel 𝑅 ∧ 𝐴 𝑅 𝐵 ) → 𝐵 ∈ ran 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		brrelex1	⊢ ( ( Rel 𝑅 ∧ 𝐴 𝑅 𝐵 ) → 𝐴 ∈ V )
2		brrelex2	⊢ ( ( Rel 𝑅 ∧ 𝐴 𝑅 𝐵 ) → 𝐵 ∈ V )
3		simpr	⊢ ( ( Rel 𝑅 ∧ 𝐴 𝑅 𝐵 ) → 𝐴 𝑅 𝐵 )
4		brelrng	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 𝑅 𝐵 ) → 𝐵 ∈ ran 𝑅 )
5	1 2 3 4	syl3anc	⊢ ( ( Rel 𝑅 ∧ 𝐴 𝑅 𝐵 ) → 𝐵 ∈ ran 𝑅 )