Theorem snopsuppss

Description: The support of a singleton containing an ordered pair is a subset of the singleton containing the first element of the ordered pair, i.e. it is empty or the singleton itself. (Contributed by AV, 19-Jul-2019)

	Ref	Expression
Assertion	snopsuppss	$$⊢\left\{⟨X,Y⟩\right\}\mathrm{supp}Z\subseteq \left\{X\right\}$$

Proof

Step	Hyp	Ref	Expression
1		suppssdm	$$⊢\left\{⟨X,Y⟩\right\}\mathrm{supp}Z\subseteq \mathrm{dom}\left\{⟨X,Y⟩\right\}$$
2		dmsnopss	$$⊢\mathrm{dom}\left\{⟨X,Y⟩\right\}\subseteq \left\{X\right\}$$
3	1 2	sstri	$$⊢\left\{⟨X,Y⟩\right\}\mathrm{supp}Z\subseteq \left\{X\right\}$$