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Metamath Proof Explorer

Theorem s3fn

Description: A length 3 word is a function with a triple as domain. (Contributed by Alexander van der Vekens, 5-Dec-2017) (Revised by AV, 23-Jan-2021)

Ref Expression
Assertion s3fn 
|- ( ( A e. V /\ B e. V /\ C e. V ) -> <" A B C "> Fn { 0 , 1 , 2 } )

Proof

Step Hyp Ref Expression
1 s3cl 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> <" A B C "> e. Word V )
2 wrdfn 
 |-  ( <" A B C "> e. Word V -> <" A B C "> Fn ( 0 ..^ ( # ` <" A B C "> ) ) )
3 1 2 syl 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> <" A B C "> Fn ( 0 ..^ ( # ` <" A B C "> ) ) )
4 s3len 
 |-  ( # ` <" A B C "> ) = 3
5 4 oveq2i 
 |-  ( 0 ..^ ( # ` <" A B C "> ) ) = ( 0 ..^ 3 )
6 fzo0to3tp 
 |-  ( 0 ..^ 3 ) = { 0 , 1 , 2 }
7 5 6 eqtr2i 
 |-  { 0 , 1 , 2 } = ( 0 ..^ ( # ` <" A B C "> ) )
8 7 fneq2i 
 |-  ( <" A B C "> Fn { 0 , 1 , 2 } <-> <" A B C "> Fn ( 0 ..^ ( # ` <" A B C "> ) ) )
9 3 8 sylibr 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> <" A B C "> Fn { 0 , 1 , 2 } )