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 ⦆ = ( { ∅ } × tag 𝐴 )
2		bj-pr1eq	⊢ ( ⦅ 𝐴 ⦆ = ( { ∅ } × tag 𝐴 ) → pr1 ⦅ 𝐴 ⦆ = pr1 ( { ∅ } × tag 𝐴 ) )
3	1 2	ax-mp	⊢ pr1 ⦅ 𝐴 ⦆ = pr1 ( { ∅ } × tag 𝐴 )
4		bj-pr1val	⊢ pr1 ( { ∅ } × tag 𝐴 ) = if ( ∅ = ∅ , 𝐴 , ∅ )
5		eqid	⊢ ∅ = ∅
6	5	iftruei	⊢ if ( ∅ = ∅ , 𝐴 , ∅ ) = 𝐴
7	3 4 6	3eqtri	⊢ pr1 ⦅ 𝐴 ⦆ = 𝐴