Theorem fnrnafv

Description: The range of a function expressed as a collection of the function's values, analogous to fnrnfv . (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	fnrnafv	$$⊢FFnA\to \mathrm{ran}F=\left\{y|\exists x\in Ay=\left(F\mathrm{\text{'}\text{'}\text{'}}x\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfafn5a	$$⊢FFnA\to F=\left(x\in A⟼\left(F\mathrm{\text{'}\text{'}\text{'}}x\right)\right)$$
2	1	rneqd	$$⊢FFnA\to \mathrm{ran}F=\mathrm{ran}\left(x\in A⟼\left(F\mathrm{\text{'}\text{'}\text{'}}x\right)\right)$$
3		eqid	$$⊢\left(x\in A⟼\left(F\mathrm{\text{'}\text{'}\text{'}}x\right)\right)=\left(x\in A⟼\left(F\mathrm{\text{'}\text{'}\text{'}}x\right)\right)$$
4	3	rnmpt	$$⊢\mathrm{ran}\left(x\in A⟼\left(F\mathrm{\text{'}\text{'}\text{'}}x\right)\right)=\left\{y|\exists x\in Ay=\left(F\mathrm{\text{'}\text{'}\text{'}}x\right)\right\}$$
5	2 4	eqtrdi	$$⊢FFnA\to \mathrm{ran}F=\left\{y|\exists x\in Ay=\left(F\mathrm{\text{'}\text{'}\text{'}}x\right)\right\}$$