Theorem s1fv

Description: Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015) (Revised by Mario Carneiro, 26-Feb-2016)

	Ref	Expression
Assertion	s1fv	⊢ ( 𝐴 ∈ 𝐵 → ( ⟨“ 𝐴 ”⟩ ‘ 0 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		s1val	⊢ ( 𝐴 ∈ 𝐵 → ⟨“ 𝐴 ”⟩ = { ⟨ 0 , 𝐴 ⟩ } )
2	1	fveq1d	⊢ ( 𝐴 ∈ 𝐵 → ( ⟨“ 𝐴 ”⟩ ‘ 0 ) = ( { ⟨ 0 , 𝐴 ⟩ } ‘ 0 ) )
3		0nn0	⊢ 0 ∈ ℕ0
4		fvsng	⊢ ( ( 0 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵 ) → ( { ⟨ 0 , 𝐴 ⟩ } ‘ 0 ) = 𝐴 )
5	3 4	mpan	⊢ ( 𝐴 ∈ 𝐵 → ( { ⟨ 0 , 𝐴 ⟩ } ‘ 0 ) = 𝐴 )
6	2 5	eqtrd	⊢ ( 𝐴 ∈ 𝐵 → ( ⟨“ 𝐴 ”⟩ ‘ 0 ) = 𝐴 )