fnovrn

Description: An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007)

		Ref	Expression
	Assertion	fnovrn	$⊢ (F Fn (A \times B) \land C \in A \land D \in B) \to C F D \in ran F$

Step	Hyp	Ref	Expression
1		opelxpi	$⊢ (C \in A \land D \in B) \to ⟨C, D⟩ \in (A \times B)$
2		df-ov	$⊢ C F D = F (⟨C, D⟩)$
3		fnfvelrn	$⊢ (F Fn (A \times B) \land ⟨C, D⟩ \in (A \times B)) \to F (⟨C, D⟩) \in ran F$
4	2 3	eqeltrid	$⊢ (F Fn (A \times B) \land ⟨C, D⟩ \in (A \times B)) \to C F D \in ran F$
5	1 4	sylan2	$⊢ (F Fn (A \times B) \land (C \in A \land D \in B)) \to C F D \in ran F$
6	5	3impb	$⊢ (F Fn (A \times B) \land C \in A \land D \in B) \to C F D \in ran F$

Metamath Proof Explorer