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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 → ⟨“ ABC ”⟩ Fn 0 1 2

Proof

Step Hyp Ref Expression
1 s3cl ⊢ A ∈ V ∧ B ∈ V ∧ C ∈ V → ⟨“ ABC ”⟩ ∈ Word V
2 wrdfn ⊢ ⟨“ ABC ”⟩ ∈ Word V → ⟨“ ABC ”⟩ Fn 0 ..^ ⟨“ ABC ”⟩
3 1 2 syl ⊢ A ∈ V ∧ B ∈ V ∧ C ∈ V → ⟨“ ABC ”⟩ Fn 0 ..^ ⟨“ ABC ”⟩
4 s3len ⊢ ⟨“ ABC ”⟩ = 3
5 4 oveq2i ⊢ 0 ..^ ⟨“ ABC ”⟩ = 0 ..^ 3
6 fzo0to3tp ⊢ 0 ..^ 3 = 0 1 2
7 5 6 eqtr2i ⊢ 0 1 2 = 0 ..^ ⟨“ ABC ”⟩
8 7 fneq2i ⊢ ⟨“ ABC ”⟩ Fn 0 1 2 ↔ ⟨“ ABC ”⟩ Fn 0 ..^ ⟨“ ABC ”⟩
9 3 8 sylibr ⊢ A ∈ V ∧ B ∈ V ∧ C ∈ V → ⟨“ ABC ”⟩ Fn 0 1 2