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Metamath Proof Explorer

Theorem indfsd

Description: The indicator function of a finite set has finite support. (Contributed by Thierry Arnoux, 18-Jan-2026)

Ref Expression
Hypotheses indfsd.1 
|- ( ph -> O e. V )
indfsd.2 
|- ( ph -> A C_ O )
indfsd.3 
|- ( ph -> A e. Fin )
Assertion indfsd 
|- ( ph -> ( ( _Ind ` O ) ` A ) finSupp 0 )

Proof

Step Hyp Ref Expression
1 indfsd.1 
 |-  ( ph -> O e. V )
2 indfsd.2 
 |-  ( ph -> A C_ O )
3 indfsd.3 
 |-  ( ph -> A e. Fin )
4 fvexd 
 |-  ( ph -> ( ( _Ind ` O ) ` A ) e. _V )
5 c0ex 
 |-  0 e. _V
6 5 a1i 
 |-  ( ph -> 0 e. _V )
7 indf 
 |-  ( ( O e. V /\ A C_ O ) -> ( ( _Ind ` O ) ` A ) : O --> { 0 , 1 } )
8 1 2 7 syl2anc 
 |-  ( ph -> ( ( _Ind ` O ) ` A ) : O --> { 0 , 1 } )
9 8 ffund 
 |-  ( ph -> Fun ( ( _Ind ` O ) ` A ) )
10 indsupp 
 |-  ( ( O e. V /\ A C_ O ) -> ( ( ( _Ind ` O ) ` A ) supp 0 ) = A )
11 1 2 10 syl2anc 
 |-  ( ph -> ( ( ( _Ind ` O ) ` A ) supp 0 ) = A )
12 11 3 eqeltrd 
 |-  ( ph -> ( ( ( _Ind ` O ) ` A ) supp 0 ) e. Fin )
13 4 6 9 12 isfsuppd 
 |-  ( ph -> ( ( _Ind ` O ) ` A ) finSupp 0 )