Theorem bj-pr2val

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

	Ref	Expression
Assertion	bj-pr2val	⊢ pr2 ( { 𝐴 } × tag 𝐵 ) = if ( 𝐴 = 1o , 𝐵 , ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-bj-pr2	⊢ pr2 ( { 𝐴 } × tag 𝐵 ) = ( 1o Proj ( { 𝐴 } × tag 𝐵 ) )
2		1oex	⊢ 1o ∈ V
3		bj-projval	⊢ ( 1o ∈ V → ( 1o Proj ( { 𝐴 } × tag 𝐵 ) ) = if ( 𝐴 = 1o , 𝐵 , ∅ ) )
4	2 3	ax-mp	⊢ ( 1o Proj ( { 𝐴 } × tag 𝐵 ) ) = if ( 𝐴 = 1o , 𝐵 , ∅ )
5	1 4	eqtri	⊢ pr2 ( { 𝐴 } × tag 𝐵 ) = if ( 𝐴 = 1o , 𝐵 , ∅ )