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Metamath Proof Explorer

Theorem funrnex

Description: If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of Monk1 p. 46. This theorem is derived using the Axiom of Replacement in the form of funex . (Contributed by NM, 11-Nov-1995)

		Ref	Expression
	Assertion	funrnex	$⊢ dom F \in B \to (Fun F \to ran F \in V)$

Proof

Step	Hyp	Ref	Expression
1		funex	$⊢ (Fun F \land dom F \in B) \to F \in V$
2	1	ex	$⊢ Fun F \to (dom F \in B \to F \in V)$
3		rnexg	$⊢ F \in V \to ran F \in V$
4	2 3	syl6com	$⊢ dom F \in B \to (Fun F \to ran F \in V)$