Metamath Proof Explorer


Theorem dprdffsupp

Description: A finitely supported function in S is a finitely supported function. (Contributed by Mario Carneiro, 25-Apr-2016) (Revised by AV, 11-Jul-2019)

Ref Expression
Hypotheses dprdff.w
|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
dprdff.1
|- ( ph -> G dom DProd S )
dprdff.2
|- ( ph -> dom S = I )
dprdff.3
|- ( ph -> F e. W )
Assertion dprdffsupp
|- ( ph -> F finSupp .0. )

Proof

Step Hyp Ref Expression
1 dprdff.w
 |-  W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
2 dprdff.1
 |-  ( ph -> G dom DProd S )
3 dprdff.2
 |-  ( ph -> dom S = I )
4 dprdff.3
 |-  ( ph -> F e. W )
5 1 2 3 dprdw
 |-  ( ph -> ( F e. W <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) ) )
6 4 5 mpbid
 |-  ( ph -> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) )
7 6 simp3d
 |-  ( ph -> F finSupp .0. )