Theorem funopafv2b

Description: Equivalence of function value and ordered pair membership, analogous to funopfvb . (Contributed by AV, 6-Sep-2022)

	Ref	Expression
Assertion	funopafv2b	$$⊢\left(\mathrm{Fun}F\wedge A\in \mathrm{dom}F\right)\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=B\leftrightarrow ⟨A,B⟩\in F\right)$$

Proof

Step	Hyp	Ref	Expression
1		funfn	$$⊢\mathrm{Fun}F\leftrightarrow FFn\mathrm{dom}F$$
2		fnopafv2b	$$⊢\left(FFn\mathrm{dom}F\wedge A\in \mathrm{dom}F\right)\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=B\leftrightarrow ⟨A,B⟩\in F\right)$$
3	1 2	sylanb	$$⊢\left(\mathrm{Fun}F\wedge A\in \mathrm{dom}F\right)\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=B\leftrightarrow ⟨A,B⟩\in F\right)$$