Metamath Proof Explorer


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 ( 𝐴𝑆 → dom ⟨“ 𝐴 ”⟩ = { 0 } )

Proof

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