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 |
|
elex |
|- ( F e. X_ i e. I ( S ` i ) -> F e. _V ) |
5 |
4
|
a1i |
|- ( ph -> ( F e. X_ i e. I ( S ` i ) -> F e. _V ) ) |
6 |
2 3
|
dprddomcld |
|- ( ph -> I e. _V ) |
7 |
|
fnex |
|- ( ( F Fn I /\ I e. _V ) -> F e. _V ) |
8 |
7
|
expcom |
|- ( I e. _V -> ( F Fn I -> F e. _V ) ) |
9 |
6 8
|
syl |
|- ( ph -> ( F Fn I -> F e. _V ) ) |
10 |
9
|
adantrd |
|- ( ph -> ( ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) -> F e. _V ) ) |
11 |
|
fveq2 |
|- ( i = x -> ( S ` i ) = ( S ` x ) ) |
12 |
11
|
cbvixpv |
|- X_ i e. I ( S ` i ) = X_ x e. I ( S ` x ) |
13 |
12
|
eleq2i |
|- ( F e. X_ i e. I ( S ` i ) <-> F e. X_ x e. I ( S ` x ) ) |
14 |
|
elixp2 |
|- ( F e. X_ x e. I ( S ` x ) <-> ( F e. _V /\ F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) |
15 |
|
3anass |
|- ( ( F e. _V /\ F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) <-> ( F e. _V /\ ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) ) |
16 |
13 14 15
|
3bitri |
|- ( F e. X_ i e. I ( S ` i ) <-> ( F e. _V /\ ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) ) |
17 |
16
|
baib |
|- ( F e. _V -> ( F e. X_ i e. I ( S ` i ) <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) ) |
18 |
17
|
a1i |
|- ( ph -> ( F e. _V -> ( F e. X_ i e. I ( S ` i ) <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) ) ) |
19 |
5 10 18
|
pm5.21ndd |
|- ( ph -> ( F e. X_ i e. I ( S ` i ) <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) ) ) |
20 |
19
|
anbi1d |
|- ( ph -> ( ( F e. X_ i e. I ( S ` i ) /\ F finSupp .0. ) <-> ( ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) /\ F finSupp .0. ) ) ) |
21 |
|
breq1 |
|- ( h = F -> ( h finSupp .0. <-> F finSupp .0. ) ) |
22 |
21 1
|
elrab2 |
|- ( F e. W <-> ( F e. X_ i e. I ( S ` i ) /\ F finSupp .0. ) ) |
23 |
|
df-3an |
|- ( ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) <-> ( ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) ) /\ F finSupp .0. ) ) |
24 |
20 22 23
|
3bitr4g |
|- ( ph -> ( F e. W <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ F finSupp .0. ) ) ) |