Theorem funopfvb

Description: Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of Monk1 p. 42. (Contributed by NM, 26-Jan-1997)

	Ref	Expression
Assertion	funopfvb	$$⊢\left(\mathrm{Fun}F\wedge A\in \mathrm{dom}F\right)\to \left(F\left(A\right)=B\leftrightarrow ⟨A,B⟩\in F\right)$$

Proof

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