Theorem bj-pr11val

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

	Ref	Expression
Assertion	bj-pr11val	$$⊢pr1⦅A⦆=A$$

Proof

Step	Hyp	Ref	Expression
1		df-bj-1upl	$$⊢⦅A⦆=\left\{\varnothing \right\}\times tagA$$
2		bj-pr1eq	$$⊢⦅A⦆=\left\{\varnothing \right\}\times tagA\to pr1⦅A⦆=pr1\left(\left\{\varnothing \right\}\times tagA\right)$$
3	1 2	ax-mp	$$⊢pr1⦅A⦆=pr1\left(\left\{\varnothing \right\}\times tagA\right)$$
4		bj-pr1val	$$⊢pr1\left(\left\{\varnothing \right\}\times tagA\right)=if\left(\varnothing =\varnothing ,A,\varnothing \right)$$
5		eqid	$$⊢\varnothing =\varnothing$$
6	5	iftruei	$$⊢if\left(\varnothing =\varnothing ,A,\varnothing \right)=A$$
7	3 4 6	3eqtri	$$⊢pr1⦅A⦆=A$$