Theorem s1dmALT

Description: Alternate version of s1dm , having a shorter proof, but requiring that A is a set. (Contributed by AV, 9-Jan-2020) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	s1dmALT	⊢ ( 𝐴 ∈ 𝑆 → dom ⟨“ 𝐴 ”⟩ = { 0 } )

Proof

Step	Hyp	Ref	Expression
1		s1val	⊢ ( 𝐴 ∈ 𝑆 → ⟨“ 𝐴 ”⟩ = { ⟨ 0 , 𝐴 ⟩ } )
2	1	dmeqd	⊢ ( 𝐴 ∈ 𝑆 → dom ⟨“ 𝐴 ”⟩ = dom { ⟨ 0 , 𝐴 ⟩ } )
3		dmsnopg	⊢ ( 𝐴 ∈ 𝑆 → dom { ⟨ 0 , 𝐴 ⟩ } = { 0 } )
4	2 3	eqtrd	⊢ ( 𝐴 ∈ 𝑆 → dom ⟨“ 𝐴 ”⟩ = { 0 } )