Theorem dmopabss

Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004)

	Ref	Expression
Assertion	dmopabss	$$⊢\mathrm{dom}\left\{⟨x,y⟩|\left(x\in A\wedge \phi \right)\right\}\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		dmopab	$$⊢\mathrm{dom}\left\{⟨x,y⟩|\left(x\in A\wedge \phi \right)\right\}=\left\{x|\exists y\left(x\in A\wedge \phi \right)\right\}$$
2		19.42v	$$⊢\exists y\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \exists y\phi \right)$$
3	2	abbii	$$⊢\left\{x|\exists y\left(x\in A\wedge \phi \right)\right\}=\left\{x|\left(x\in A\wedge \exists y\phi \right)\right\}$$
4		ssab2	$$⊢\left\{x|\left(x\in A\wedge \exists y\phi \right)\right\}\subseteq A$$
5	3 4	eqsstri	$$⊢\left\{x|\exists y\left(x\in A\wedge \phi \right)\right\}\subseteq A$$
6	1 5	eqsstri	$$⊢\mathrm{dom}\left\{⟨x,y⟩|\left(x\in A\wedge \phi \right)\right\}\subseteq A$$