Theorem funbrfvb

Description: Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006)

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

Proof

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