Theorem fsuppmptdm

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

	Ref	Expression
Hypotheses	fsuppmptdm.f	|- F = ( x e. A |-> Y )
	fsuppmptdm.a	|- ( ph -> A e. Fin )
	fsuppmptdm.y	|- ( ( ph /\ x e. A ) -> Y e. V )
	fsuppmptdm.z	|- ( ph -> Z e. W )
Assertion	fsuppmptdm	|- ( ph -> F finSupp Z )

Proof

Step	Hyp	Ref	Expression
1		fsuppmptdm.f	|- F = ( x e. A |-> Y )
2		fsuppmptdm.a	|- ( ph -> A e. Fin )
3		fsuppmptdm.y	|- ( ( ph /\ x e. A ) -> Y e. V )
4		fsuppmptdm.z	|- ( ph -> Z e. W )
5	3 1	fmptd	|- ( ph -> F : A --> V )
6	5 2 4	fdmfifsupp	|- ( ph -> F finSupp Z )