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Metamath Proof Explorer

Theorem indfsid

Description: Conditions for a function to be an indicator function. (Contributed by Thierry Arnoux, 18-Jan-2026)

Ref Expression
Hypotheses indfsid.1 
|- ( ph -> O e. V )
indfsid.2 
|- ( ph -> F : O --> { 0 , 1 } )
Assertion indfsid 
|- ( ph -> F = ( ( _Ind ` O ) ` ( F supp 0 ) ) )

Proof

Step Hyp Ref Expression
1 indfsid.1 
 |-  ( ph -> O e. V )
2 indfsid.2 
 |-  ( ph -> F : O --> { 0 , 1 } )
3 indpreima 
 |-  ( ( O e. V /\ F : O --> { 0 , 1 } ) -> F = ( ( _Ind ` O ) ` ( `' F " { 1 } ) ) )
4 1 2 3 syl2anc 
 |-  ( ph -> F = ( ( _Ind ` O ) ` ( `' F " { 1 } ) ) )
5 c0ex 
 |-  0 e. _V
6 5 a1i 
 |-  ( ph -> 0 e. _V )
7 fsuppeq 
 |-  ( ( O e. V /\ 0 e. _V ) -> ( F : O --> { 0 , 1 } -> ( F supp 0 ) = ( `' F " ( { 0 , 1 } \ { 0 } ) ) ) )
8 7 imp 
 |-  ( ( ( O e. V /\ 0 e. _V ) /\ F : O --> { 0 , 1 } ) -> ( F supp 0 ) = ( `' F " ( { 0 , 1 } \ { 0 } ) ) )
9 1 6 2 8 syl21anc 
 |-  ( ph -> ( F supp 0 ) = ( `' F " ( { 0 , 1 } \ { 0 } ) ) )
10 0ne1 
 |-  0 =/= 1
11 difprsn1 
 |-  ( 0 =/= 1 -> ( { 0 , 1 } \ { 0 } ) = { 1 } )
12 10 11 mp1i 
 |-  ( ph -> ( { 0 , 1 } \ { 0 } ) = { 1 } )
13 12 imaeq2d 
 |-  ( ph -> ( `' F " ( { 0 , 1 } \ { 0 } ) ) = ( `' F " { 1 } ) )
14 9 13 eqtrd 
 |-  ( ph -> ( F supp 0 ) = ( `' F " { 1 } ) )
15 14 fveq2d 
 |-  ( ph -> ( ( _Ind ` O ) ` ( F supp 0 ) ) = ( ( _Ind ` O ) ` ( `' F " { 1 } ) ) )
16 4 15 eqtr4d 
 |-  ( ph -> F = ( ( _Ind ` O ) ` ( F supp 0 ) ) )