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Metamath Proof Explorer

Theorem rnexg

Description: The range of a set is a set. Corollary 6.8(3) of TakeutiZaring p. 26. Similar to Lemma 3D of Enderton p. 41. (Contributed by NM, 31-Mar-1995)

		Ref	Expression
	Assertion	rnexg	$⊢ A \in V \to ran A \in V$

Proof

Step	Hyp	Ref	Expression
1		uniexg	$⊢ A \in V \to ⋃ A \in V$
2		uniexg	$⊢ ⋃ A \in V \to ⋃ ⋃ A \in V$
3		ssun2	$⊢ ran A \subseteq dom A \cup ran A$
4		dmrnssfld	$⊢ dom A \cup ran A \subseteq ⋃ ⋃ A$
5	3 4	sstri	$⊢ ran A \subseteq ⋃ ⋃ A$
6		ssexg	$⊢ (ran A \subseteq ⋃ ⋃ A \land ⋃ ⋃ A \in V) \to ran A \in V$
7	5 6	mpan	$⊢ ⋃ ⋃ A \in V \to ran A \in V$
8	1 2 7	3syl	$⊢ A \in V \to ran A \in V$