Metamath Proof Explorer


Theorem dprdff

Description: A finitely supported function in S is a function into the base. (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 )
dprdff.b
|- B = ( Base ` G )
Assertion dprdff
|- ( ph -> F : I --> B )

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 dprdff.b
 |-  B = ( Base ` G )
6 1 2 3 dprdw
 |-  ( ph -> ( F e. W <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) ) )
7 4 6 mpbid
 |-  ( ph -> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) )
8 7 simp1d
 |-  ( ph -> F Fn I )
9 7 simp2d
 |-  ( ph -> A. x e. I ( F ` x ) e. ( S ` x ) )
10 2 3 dprdf2
 |-  ( ph -> S : I --> ( SubGrp ` G ) )
11 10 ffvelrnda
 |-  ( ( ph /\ x e. I ) -> ( S ` x ) e. ( SubGrp ` G ) )
12 5 subgss
 |-  ( ( S ` x ) e. ( SubGrp ` G ) -> ( S ` x ) C_ B )
13 11 12 syl
 |-  ( ( ph /\ x e. I ) -> ( S ` x ) C_ B )
14 13 sseld
 |-  ( ( ph /\ x e. I ) -> ( ( F ` x ) e. ( S ` x ) -> ( F ` x ) e. B ) )
15 14 ralimdva
 |-  ( ph -> ( A. x e. I ( F ` x ) e. ( S ` x ) -> A. x e. I ( F ` x ) e. B ) )
16 9 15 mpd
 |-  ( ph -> A. x e. I ( F ` x ) e. B )
17 ffnfv
 |-  ( F : I --> B <-> ( F Fn I /\ A. x e. I ( F ` x ) e. B ) )
18 8 16 17 sylanbrc
 |-  ( ph -> F : I --> B )