Theorem bj-opabssvv

Description: A variant of relopabiv (which could be proved from it, similarly to relxp from xpss ). (Contributed by BJ, 28-Dec-2023)

	Ref	Expression
Assertion	bj-opabssvv	$$⊢\left\{⟨x,y⟩|\phi \right\}\subseteq \mathrm{V}\times \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		vex	$$⊢y\in \mathrm{V}$$
3	1 2	pm3.2i	$$⊢\left(x\in \mathrm{V}\wedge y\in \mathrm{V}\right)$$
4	3	a1i	$$⊢\phi \to \left(x\in \mathrm{V}\wedge y\in \mathrm{V}\right)$$
5	4	ssopab2i	$$⊢\left\{⟨x,y⟩|\phi \right\}\subseteq \left\{⟨x,y⟩|\left(x\in \mathrm{V}\wedge y\in \mathrm{V}\right)\right\}$$
6		df-xp	$$⊢\mathrm{V}\times \mathrm{V}=\left\{⟨x,y⟩|\left(x\in \mathrm{V}\wedge y\in \mathrm{V}\right)\right\}$$
7	5 6	sseqtrri	$$⊢\left\{⟨x,y⟩|\phi \right\}\subseteq \mathrm{V}\times \mathrm{V}$$