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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ⟨“ 𝐴 𝐵 𝐶 ”⟩ Fn { 0 , 1 , 2 } )

Proof

Step	Hyp	Ref	Expression
1		s3cl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word 𝑉 )
2		wrdfn	⊢ ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word 𝑉 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ Fn ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ) ) )
3	1 2	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ⟨“ 𝐴 𝐵 𝐶 ”⟩ Fn { 0 , 1 , 2 } )