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Metamath Proof Explorer

Theorem fnrnfv

Description: The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005) (Proof shortened by Mario Carneiro, 31-Aug-2015)

		Ref	Expression
	Assertion	fnrnfv	$⊢ F Fn A \to ran F = \{y \| \exists x \in A y = F (x)\}$

Proof

Step	Hyp	Ref	Expression
1		dffn5	$⊢ F Fn A \leftrightarrow F = (x \in A ⟼ F (x))$
2		rneq	$⊢ F = (x \in A ⟼ F (x)) \to ran F = ran (x \in A ⟼ F (x))$
3	1 2	sylbi	$⊢ F Fn A \to ran F = ran (x \in A ⟼ F (x))$
4		eqid	$⊢ (x \in A ⟼ F (x)) = (x \in A ⟼ F (x))$
5	4	rnmpt	$⊢ ran (x \in A ⟼ F (x)) = \{y \| \exists x \in A y = F (x)\}$
6	3 5	eqtrdi	$⊢ F Fn A \to ran F = \{y \| \exists x \in A y = F (x)\}$