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	$$⊢A\in S\to \mathrm{dom}⟨“A”⟩=\left\{0\right\}$$

Proof

Step	Hyp	Ref	Expression
1		s1val	$$⊢A\in S\to ⟨“A”⟩=\left\{⟨0,A⟩\right\}$$
2	1	dmeqd	$$⊢A\in S\to \mathrm{dom}⟨“A”⟩=\mathrm{dom}\left\{⟨0,A⟩\right\}$$
3		dmsnopg	$$⊢A\in S\to \mathrm{dom}\left\{⟨0,A⟩\right\}=\left\{0\right\}$$
4	2 3	eqtrd	$$⊢A\in S\to \mathrm{dom}⟨“A”⟩=\left\{0\right\}$$