Theorem bj-pr1val

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

	Ref	Expression
Assertion	bj-pr1val	⊢ pr1 ( { 𝐴 } × tag 𝐵 ) = if ( 𝐴 = ∅ , 𝐵 , ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-bj-pr1	⊢ pr1 ( { 𝐴 } × tag 𝐵 ) = ( ∅ Proj ( { 𝐴 } × tag 𝐵 ) )
2		0ex	⊢ ∅ ∈ V
3		bj-projval	⊢ ( ∅ ∈ V → ( ∅ Proj ( { 𝐴 } × tag 𝐵 ) ) = if ( 𝐴 = ∅ , 𝐵 , ∅ ) )
4	2 3	ax-mp	⊢ ( ∅ Proj ( { 𝐴 } × tag 𝐵 ) ) = if ( 𝐴 = ∅ , 𝐵 , ∅ )
5	1 4	eqtri	⊢ pr1 ( { 𝐴 } × tag 𝐵 ) = if ( 𝐴 = ∅ , 𝐵 , ∅ )