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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 ∈ 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 } )