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Metamath Proof Explorer

Theorem relssdmrn

Description: A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of Mendelson p. 235. (Contributed by NM, 3-Aug-1994) (Proof shortened by SN, 23-Dec-2024)

		Ref	Expression
	Assertion	relssdmrn	$⊢ Rel A \to A \subseteq dom A \times ran A$

Proof

Step	Hyp	Ref	Expression
1		id	$⊢ Rel A \to Rel A$
2		vex	$⊢ x \in V$
3		vex	$⊢ y \in V$
4	2 3	opeldm	$⊢ ⟨x, y⟩ \in A \to x \in dom A$
5	2 3	opelrn	$⊢ ⟨x, y⟩ \in A \to y \in ran A$
6	4 5	opelxpd	$⊢ ⟨x, y⟩ \in A \to ⟨x, y⟩ \in (dom A \times ran A)$
7	6	a1i	$⊢ Rel A \to (⟨x, y⟩ \in A \to ⟨x, y⟩ \in (dom A \times ran A))$
8	1 7	relssdv	$⊢ Rel A \to A \subseteq dom A \times ran A$