Theorem s2dmALT

Description: Alternate version of s2dm , having a shorter proof, but requiring that A and B are sets. (Contributed by AV, 9-Jan-2020) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	s2dmALT	$$⊢\left(A\in S\wedge B\in S\right)\to \mathrm{dom}⟨“AB”⟩=\left\{0,1\right\}$$

Proof

Step	Hyp	Ref	Expression
1		s2prop	$$⊢\left(A\in S\wedge B\in S\right)\to ⟨“AB”⟩=\left\{⟨0,A⟩,⟨1,B⟩\right\}$$
2	1	dmeqd	$$⊢\left(A\in S\wedge B\in S\right)\to \mathrm{dom}⟨“AB”⟩=\mathrm{dom}\left\{⟨0,A⟩,⟨1,B⟩\right\}$$
3		dmpropg	$$⊢\left(A\in S\wedge B\in S\right)\to \mathrm{dom}\left\{⟨0,A⟩,⟨1,B⟩\right\}=\left\{0,1\right\}$$
4	2 3	eqtrd	$$⊢\left(A\in S\wedge B\in S\right)\to \mathrm{dom}⟨“AB”⟩=\left\{0,1\right\}$$