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Metamath Proof Explorer

Theorem dprdw

Description: The property of being a finitely supported function in the family S . (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 )
Assertion dprdw 
|- ( ph -> ( F e. W <-> ( F Fn I /\ A. x e. I ( F ` x ) e. ( S ` x ) /\ 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 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. ) ) )