Metamath Proof Explorer

Theorem relssdmrnOLD

Description: Obsolete version of relssdmrn as of 23-Dec-2024. (Contributed by NM, 3-Aug-1994) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		id	$⊢ Rel A \to Rel A$
2		19.8a	$⊢ ⟨x, y⟩ \in A \to \exists y ⟨x, y⟩ \in A$
3		19.8a	$⊢ ⟨x, y⟩ \in A \to \exists x ⟨x, y⟩ \in A$
4		opelxp	$⊢ ⟨x, y⟩ \in (dom A \times ran A) \leftrightarrow (x \in dom A \land y \in ran A)$
5		vex	$⊢ x \in V$
6	5	eldm2	$⊢ x \in dom A \leftrightarrow \exists y ⟨x, y⟩ \in A$
7		vex	$⊢ y \in V$
8	7	elrn2	$⊢ y \in ran A \leftrightarrow \exists x ⟨x, y⟩ \in A$
9	6 8	anbi12i	$⊢ (x \in dom A \land y \in ran A) \leftrightarrow (\exists y ⟨x, y⟩ \in A \land \exists x ⟨x, y⟩ \in A)$
10	4 9	bitri	$⊢ ⟨x, y⟩ \in (dom A \times ran A) \leftrightarrow (\exists y ⟨x, y⟩ \in A \land \exists x ⟨x, y⟩ \in A)$
11	2 3 10	sylanbrc	$⊢ ⟨x, y⟩ \in A \to ⟨x, y⟩ \in (dom A \times ran A)$
12	11	a1i	$⊢ Rel A \to (⟨x, y⟩ \in A \to ⟨x, y⟩ \in (dom A \times ran A))$
13	1 12	relssdv	$⊢ Rel A \to A \subseteq dom A \times ran A$