Metamath Proof Explorer


Theorem fidmfisupp

Description: A function with a finite domain is finitely supported. (Contributed by Glauco Siliprandi, 24-Dec-2020)

Ref Expression
Hypotheses fidmfisupp.1
|- ( ph -> F : D --> R )
fidmfisupp.2
|- ( ph -> D e. Fin )
fidmfisupp.3
|- ( ph -> Z e. V )
Assertion fidmfisupp
|- ( ph -> F finSupp Z )

Proof

Step Hyp Ref Expression
1 fidmfisupp.1
 |-  ( ph -> F : D --> R )
2 fidmfisupp.2
 |-  ( ph -> D e. Fin )
3 fidmfisupp.3
 |-  ( ph -> Z e. V )
4 1 2 fexd
 |-  ( ph -> F e. _V )
5 suppimacnv
 |-  ( ( F e. _V /\ Z e. V ) -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
6 4 3 5 syl2anc
 |-  ( ph -> ( F supp Z ) = ( `' F " ( _V \ { Z } ) ) )
7 2 1 fisuppfi
 |-  ( ph -> ( `' F " ( _V \ { Z } ) ) e. Fin )
8 6 7 eqeltrd
 |-  ( ph -> ( F supp Z ) e. Fin )
9 1 ffund
 |-  ( ph -> Fun F )
10 funisfsupp
 |-  ( ( Fun F /\ F e. _V /\ Z e. V ) -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
11 9 4 3 10 syl3anc
 |-  ( ph -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
12 8 11 mpbird
 |-  ( ph -> F finSupp Z )