Theorem fsuppmptdm

Description: A mapping with a finite domain is finitely supported. (Contributed by AV, 7-Jun-2019)

	Ref	Expression
Hypotheses	fsuppmptdm.f	$$⊢F=\left(x\in A⟼Y\right)$$
	fsuppmptdm.a	$$⊢\phi \to A\in \mathrm{Fin}$$
	fsuppmptdm.y	$$⊢\left(\phi \wedge x\in A\right)\to Y\in V$$
	fsuppmptdm.z	$$⊢\phi \to Z\in W$$
Assertion	fsuppmptdm	$$⊢\phi \to {finSupp}_Z\left(F\right)$$

Proof

Step	Hyp	Ref	Expression
1		fsuppmptdm.f	$$⊢F=\left(x\in A⟼Y\right)$$
2		fsuppmptdm.a	$$⊢\phi \to A\in \mathrm{Fin}$$
3		fsuppmptdm.y	$$⊢\left(\phi \wedge x\in A\right)\to Y\in V$$
4		fsuppmptdm.z	$$⊢\phi \to Z\in W$$
5	3 1	fmptd	$$⊢\phi \to F:A⟶V$$
6	5 2 4	fdmfifsupp	$$⊢\phi \to {finSupp}_Z\left(F\right)$$