Theorem dmxpss

Description: The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007)

	Ref	Expression
Assertion	dmxpss	$$⊢\mathrm{dom}\left(A\times B\right)\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		xpeq2	$$⊢B=\varnothing \to A\times B=A\times \varnothing$$
2		xp0	$$⊢A\times \varnothing =\varnothing$$
3	1 2	eqtrdi	$$⊢B=\varnothing \to A\times B=\varnothing$$
4	3	dmeqd	$$⊢B=\varnothing \to \mathrm{dom}\left(A\times B\right)=\mathrm{dom}\varnothing$$
5		dm0	$$⊢\mathrm{dom}\varnothing =\varnothing$$
6	4 5	eqtrdi	$$⊢B=\varnothing \to \mathrm{dom}\left(A\times B\right)=\varnothing$$
7		0ss	$$⊢\varnothing \subseteq A$$
8	6 7	eqsstrdi	$$⊢B=\varnothing \to \mathrm{dom}\left(A\times B\right)\subseteq A$$
9		dmxp	$$⊢B\ne \varnothing \to \mathrm{dom}\left(A\times B\right)=A$$
10		eqimss	$$⊢\mathrm{dom}\left(A\times B\right)=A\to \mathrm{dom}\left(A\times B\right)\subseteq A$$
11	9 10	syl	$$⊢B\ne \varnothing \to \mathrm{dom}\left(A\times B\right)\subseteq A$$
12	8 11	pm2.61ine	$$⊢\mathrm{dom}\left(A\times B\right)\subseteq A$$