Theorem bj-pr2val

Description: Value of the second projection. (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-pr2val	$$⊢pr2\left(\left\{A\right\}\times tagB\right)=if\left(A=1_{𝑜},B,\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-bj-pr2	$$⊢pr2\left(\left\{A\right\}\times tagB\right)=\left(1_{𝑜}Proj\left(\left\{A\right\}\times tagB\right)\right)$$
2		1oex	$$⊢1_{𝑜}\in \mathrm{V}$$
3		bj-projval	$$⊢1_{𝑜}\in \mathrm{V}\to \left(1_{𝑜}Proj\left(\left\{A\right\}\times tagB\right)\right)=if\left(A=1_{𝑜},B,\varnothing \right)$$
4	2 3	ax-mp	$$⊢\left(1_{𝑜}Proj\left(\left\{A\right\}\times tagB\right)\right)=if\left(A=1_{𝑜},B,\varnothing \right)$$
5	1 4	eqtri	$$⊢pr2\left(\left\{A\right\}\times tagB\right)=if\left(A=1_{𝑜},B,\varnothing \right)$$