Theorem afv20fv0

Description: If the alternate function value at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	afv20fv0	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \to F\left(A\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		afv20defat	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \to F\mathrm{defAt}A$$
2		dfatafv2eqfv	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$
3	2	eqcomd	$$⊢F\mathrm{defAt}A\to F\left(A\right)=\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)$$
4	3	adantr	$$⊢\left(F\mathrm{defAt}A\wedge \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \right)\to F\left(A\right)=\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)$$
5		simpr	$$⊢\left(F\mathrm{defAt}A\wedge \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \right)\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing$$
6	4 5	eqtrd	$$⊢\left(F\mathrm{defAt}A\wedge \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \right)\to F\left(A\right)=\varnothing$$
7	1 6	mpancom	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \to F\left(A\right)=\varnothing$$