Theorem afv2fv0b

Description: The function's value at an argument is the empty set if and only if the alternate function value at this argument is the empty set or undefined. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	afv2fv0b	$$⊢F\left(A\right)=\varnothing \leftrightarrow \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \vee \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\right)$$

Proof

Step	Hyp	Ref	Expression
1		afv2fv0	$$⊢F\left(A\right)=\varnothing \to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \vee \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\right)$$
2		afv20fv0	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \to F\left(A\right)=\varnothing$$
3		afv2ndeffv0	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\to F\left(A\right)=\varnothing$$
4	2 3	jaoi	$$⊢\left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \vee \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\right)\to F\left(A\right)=\varnothing$$
5	1 4	impbii	$$⊢F\left(A\right)=\varnothing \leftrightarrow \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \vee \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\right)$$