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 e. ( `' A " { B } ) <-> C A B ) )

Proof

Step	Hyp	Ref	Expression
1		relcnv	|- Rel `' A
2		elrelimasn	|- ( Rel `' A -> ( C e. ( `' A " { B } ) <-> B `' A C ) )
3	1 2	ax-mp	|- ( C e. ( `' A " { B } ) <-> B `' A C )
4		relbrcnvg	|- ( Rel A -> ( B `' A C <-> C A B ) )
5	3 4	syl5bb	|- ( Rel A -> ( C e. ( `' A " { B } ) <-> C A B ) )