Theorem fsuppmptdmf

Description: A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019)

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

Proof

Step	Hyp	Ref	Expression
1		fsuppmptdmf.n	|- F/ x ph
2		fsuppmptdmf.f	|- F = ( x e. A |-> Y )
3		fsuppmptdmf.a	|- ( ph -> A e. Fin )
4		fsuppmptdmf.y	|- ( ( ph /\ x e. A ) -> Y e. V )
5		fsuppmptdmf.z	|- ( ph -> Z e. W )
6	1 4 2	fmptdf	|- ( ph -> F : A --> V )
7	6 3 5	fdmfifsupp	|- ( ph -> F finSupp Z )