Theorem eliniseg2

Description: Eliminate the class existence constraint in eliniseg . (Contributed by Mario Carneiro, 5-Dec-2014) (Revised by Mario Carneiro, 17-Nov-2015)

	Ref	Expression
Assertion	eliniseg2	⊢ ( Rel 𝐴 → ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝐶 𝐴 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		relcnv	⊢ Rel ◡ 𝐴
2		elrelimasn	⊢ ( Rel ◡ 𝐴 → ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝐵 ◡ 𝐴 𝐶 ) )
3	1 2	ax-mp	⊢ ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝐵 ◡ 𝐴 𝐶 )
4		relbrcnvg	⊢ ( Rel 𝐴 → ( 𝐵 ◡ 𝐴 𝐶 ↔ 𝐶 𝐴 𝐵 ) )
5	3 4	bitrid	⊢ ( Rel 𝐴 → ( 𝐶 ∈ ( ◡ 𝐴 “ { 𝐵 } ) ↔ 𝐶 𝐴 𝐵 ) )