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 } )