Theorem funbrafvb

Description: Equivalence of function value and binary relation, analogous to funbrfvb . (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	funbrafvb	$$⊢\left(\mathrm{Fun}F\wedge A\in \mathrm{dom}F\right)\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=B\leftrightarrow AFB\right)$$

Proof

Step	Hyp	Ref	Expression
1		funfn	$$⊢\mathrm{Fun}F\leftrightarrow FFn\mathrm{dom}F$$
2		fnbrafvb	$$⊢\left(FFn\mathrm{dom}F\wedge A\in \mathrm{dom}F\right)\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=B\leftrightarrow AFB\right)$$
3	1 2	sylanb	$$⊢\left(\mathrm{Fun}F\wedge A\in \mathrm{dom}F\right)\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=B\leftrightarrow AFB\right)$$