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Metamath Proof Explorer

Theorem finnzfsuppd

Description: If a function is zero outside of a finite set, it has finite support. (Contributed by Rohan Ridenour, 13-May-2024)

Ref Expression
Hypotheses finnzfsuppd.1 
|- ( ph -> F e. V )
finnzfsuppd.2 
|- ( ph -> F Fn D )
finnzfsuppd.3 
|- ( ph -> Z e. U )
finnzfsuppd.4 
|- ( ph -> A e. Fin )
finnzfsuppd.5 
|- ( ( ph /\ x e. D ) -> ( x e. A \/ ( F ` x ) = Z ) )
Assertion finnzfsuppd 
|- ( ph -> F finSupp Z )

Proof

Step Hyp Ref Expression
1 finnzfsuppd.1 
 |-  ( ph -> F e. V )
2 finnzfsuppd.2 
 |-  ( ph -> F Fn D )
3 finnzfsuppd.3 
 |-  ( ph -> Z e. U )
4 finnzfsuppd.4 
 |-  ( ph -> A e. Fin )
5 finnzfsuppd.5 
 |-  ( ( ph /\ x e. D ) -> ( x e. A \/ ( F ` x ) = Z ) )
6 1 2 fndmexd 
 |-  ( ph -> D e. _V )
7 elsuppfn 
 |-  ( ( F Fn D /\ D e. _V /\ Z e. U ) -> ( x e. ( F supp Z ) <-> ( x e. D /\ ( F ` x ) =/= Z ) ) )
8 2 6 3 7 syl3anc 
 |-  ( ph -> ( x e. ( F supp Z ) <-> ( x e. D /\ ( F ` x ) =/= Z ) ) )
9 8 biimpa 
 |-  ( ( ph /\ x e. ( F supp Z ) ) -> ( x e. D /\ ( F ` x ) =/= Z ) )
10 9 simpld 
 |-  ( ( ph /\ x e. ( F supp Z ) ) -> x e. D )
11 10 5 syldan 
 |-  ( ( ph /\ x e. ( F supp Z ) ) -> ( x e. A \/ ( F ` x ) = Z ) )
12 9 simprd 
 |-  ( ( ph /\ x e. ( F supp Z ) ) -> ( F ` x ) =/= Z )
13 12 neneqd 
 |-  ( ( ph /\ x e. ( F supp Z ) ) -> -. ( F ` x ) = Z )
14 11 13 olcnd 
 |-  ( ( ph /\ x e. ( F supp Z ) ) -> x e. A )
15 14 ex 
 |-  ( ph -> ( x e. ( F supp Z ) -> x e. A ) )
16 15 ssrdv 
 |-  ( ph -> ( F supp Z ) C_ A )
17 4 16 ssfid 
 |-  ( ph -> ( F supp Z ) e. Fin )
18 fnfun 
 |-  ( F Fn D -> Fun F )
19 2 18 syl 
 |-  ( ph -> Fun F )
20 funisfsupp 
 |-  ( ( Fun F /\ F e. V /\ Z e. U ) -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
21 19 1 3 20 syl3anc 
 |-  ( ph -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
22 17 21 mpbird 
 |-  ( ph -> F finSupp Z )