Theorem s2dm

Description: The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020)

	Ref	Expression
Assertion	s2dm	⊢ dom ⟨“ 𝐴 𝐵 ”⟩ = { 0 , 1 }

Proof

Step	Hyp	Ref	Expression
1		s2cli	⊢ ⟨“ 𝐴 𝐵 ”⟩ ∈ Word V
2		wrdf	⊢ ( ⟨“ 𝐴 𝐵 ”⟩ ∈ Word V → ⟨“ 𝐴 𝐵 ”⟩ : ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) ) ⟶ V )
3	1 2	ax-mp	⊢ ⟨“ 𝐴 𝐵 ”⟩ : ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) ) ⟶ V
4		s2len	⊢ ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) = 2
5		oveq2	⊢ ( ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) = 2 → ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) ) = ( 0 ..^ 2 ) )
6		fzo0to2pr	⊢ ( 0 ..^ 2 ) = { 0 , 1 }
7	5 6	eqtrdi	⊢ ( ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) = 2 → ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) ) = { 0 , 1 } )
8	4 7	ax-mp	⊢ ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) ) = { 0 , 1 }
9	8	feq2i	⊢ ( ⟨“ 𝐴 𝐵 ”⟩ : ( 0 ..^ ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) ) ⟶ V ↔ ⟨“ 𝐴 𝐵 ”⟩ : { 0 , 1 } ⟶ V )
10	3 9	mpbi	⊢ ⟨“ 𝐴 𝐵 ”⟩ : { 0 , 1 } ⟶ V
11	10	fdmi	⊢ dom ⟨“ 𝐴 𝐵 ”⟩ = { 0 , 1 }