Metamath Proof Explorer

Theorem funex

Description: If the domain of a function exists, so does the function. Part of Theorem 4.15(v) of Monk1 p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex . (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995)

		Ref	Expression
	Assertion	funex	$⊢ (Fun F \land dom F \in B) \to F \in V$

Proof

Step	Hyp	Ref	Expression
1		funfn	$⊢ Fun F \leftrightarrow F Fn dom F$
2		fnex	$⊢ (F Fn dom F \land dom F \in B) \to F \in V$
3	1 2	sylanb	$⊢ (Fun F \land dom F \in B) \to F \in V$