This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 2ndrn

Description: The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006)

		Ref	Expression
	Assertion	2ndrn	$⊢ (Rel R \land A \in R) \to 2^{nd} (A) \in ran R$

Step	Hyp	Ref	Expression
1		1st2nd	$⊢ (Rel R \land A \in R) \to A = ⟨1^{st} (A), 2^{nd} (A)⟩$
2		simpr	$⊢ (Rel R \land A \in R) \to A \in R$
3	1 2	eqeltrrd	$⊢ (Rel R \land A \in R) \to ⟨1^{st} (A), 2^{nd} (A)⟩ \in R$
4		fvex	$⊢ 1^{st} (A) \in V$
5		fvex	$⊢ 2^{nd} (A) \in V$
6	4 5	opelrn	$⊢ ⟨1^{st} (A), 2^{nd} (A)⟩ \in R \to 2^{nd} (A) \in ran R$
7	3 6	syl	$⊢ (Rel R \land A \in R) \to 2^{nd} (A) \in ran R$