Theorem fpwfvss

Description: Functions into a powerset always have values which are subsets. This is dependant on our convention when the argument is not part of the domain. (Contributed by RP, 13-Sep-2024)

	Ref	Expression
Hypothesis	fpwfvss.f	$$⊢F:C⟶𝒫B$$
Assertion	fpwfvss	$$⊢F\left(A\right)\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		fpwfvss.f	$$⊢F:C⟶𝒫B$$
2	1	ffvelcdmi	$$⊢A\in C\to F\left(A\right)\in 𝒫B$$
3	2	elpwid	$$⊢A\in C\to F\left(A\right)\subseteq B$$
4	1	fdmi	$$⊢\mathrm{dom}F=C$$
5	4	eleq2i	$$⊢A\in \mathrm{dom}F\leftrightarrow A\in C$$
6		ndmfv	$$⊢\neg A\in \mathrm{dom}F\to F\left(A\right)=\varnothing$$
7	5 6	sylnbir	$$⊢\neg A\in C\to F\left(A\right)=\varnothing$$
8		0ss	$$⊢\varnothing \subseteq B$$
9	7 8	eqsstrdi	$$⊢\neg A\in C\to F\left(A\right)\subseteq B$$
10	3 9	pm2.61i	$$⊢F\left(A\right)\subseteq B$$