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Metamath Proof Explorer

Theorem s1dm

Description: The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020)

Ref Expression
Assertion s1dm dom ⟨“ 𝐴 ”⟩ = { 0 }

Proof

Step Hyp Ref Expression
1 s1cli ⊢ ⟨“ 𝐴 ”⟩ ∈ Word V
2 wrdf ⊢ ( ⟨“ 𝐴 ”⟩ ∈ Word V → ⟨“ 𝐴 ”⟩ : ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) ) ⟶ V )
3 1 2 ax-mp ⊢ ⟨“ 𝐴 ”⟩ : ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) ) ⟶ V
4 s1len ⊢ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) = 1
5 oveq2 ⊢ ( ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) = 1 → ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) ) = ( 0 ..^ 1 ) )
6 fzo01 ⊢ ( 0 ..^ 1 ) = { 0 }
7 5 6 eqtrdi ⊢ ( ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) = 1 → ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) ) = { 0 } )
8 4 7 ax-mp ⊢ ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) ) = { 0 }
9 8 eqcomi ⊢ { 0 } = ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) )
10 9 feq2i ⊢ ( ⟨“ 𝐴 ”⟩ : { 0 } ⟶ V ↔ ⟨“ 𝐴 ”⟩ : ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) ) ⟶ V )
11 3 10 mpbir ⊢ ⟨“ 𝐴 ”⟩ : { 0 } ⟶ V
12 11 fdmi ⊢ dom ⟨“ 𝐴 ”⟩ = { 0 }