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. ) |
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. ) |