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	$$⊢FFnA\to \mathrm{ran}F=\left\{y|\exists x\in Ay=F\left(x\right)\right\}$$

Proof

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