Theorem dvdszrcl

Description: Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015)

	Ref	Expression
Assertion	dvdszrcl	|- ( X || Y -> ( X e. ZZ /\ Y e. ZZ ) )

Proof

Step	Hyp	Ref	Expression
1		df-dvds	|- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. z e. ZZ ( z x. x ) = y ) }
2		opabssxp	|- { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. z e. ZZ ( z x. x ) = y ) } C_ ( ZZ X. ZZ )
3	1 2	eqsstri	|- || C_ ( ZZ X. ZZ )
4	3	brel	|- ( X || Y -> ( X e. ZZ /\ Y e. ZZ ) )