Theorem predun

Description: Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011)

	Ref	Expression
Assertion	predun	|- Pred ( R , ( A u. B ) , X ) = ( Pred ( R , A , X ) u. Pred ( R , B , X ) )

Proof

Step	Hyp	Ref	Expression
1		indir	|- ( ( A u. B ) i^i ( `' R " { X } ) ) = ( ( A i^i ( `' R " { X } ) ) u. ( B i^i ( `' R " { X } ) ) )
2		df-pred	|- Pred ( R , ( A u. B ) , X ) = ( ( A u. B ) i^i ( `' R " { X } ) )
3		df-pred	|- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) )
4		df-pred	|- Pred ( R , B , X ) = ( B i^i ( `' R " { X } ) )
5	3 4	uneq12i	|- ( Pred ( R , A , X ) u. Pred ( R , B , X ) ) = ( ( A i^i ( `' R " { X } ) ) u. ( B i^i ( `' R " { X } ) ) )
6	1 2 5	3eqtr4i	|- Pred ( R , ( A u. B ) , X ) = ( Pred ( R , A , X ) u. Pred ( R , B , X ) )