Theorem dfxp3

Description: Define the Cartesian product of three classes. Compare df-xp . (Contributed by FL, 6-Nov-2013) (Proof shortened by Mario Carneiro, 3-Nov-2015)

	Ref	Expression
Assertion	dfxp3	|- ( ( A X. B ) X. C ) = { <. <. x , y >. , z >. | ( x e. A /\ y e. B /\ z e. C ) }

Proof

Step	Hyp	Ref	Expression
1		biidd	|- ( u = <. x , y >. -> ( z e. C <-> z e. C ) )
2	1	dfoprab4	|- { <. u , z >. | ( u e. ( A X. B ) /\ z e. C ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z e. C ) }
3		df-xp	|- ( ( A X. B ) X. C ) = { <. u , z >. | ( u e. ( A X. B ) /\ z e. C ) }
4		df-3an	|- ( ( x e. A /\ y e. B /\ z e. C ) <-> ( ( x e. A /\ y e. B ) /\ z e. C ) )
5	4	oprabbii	|- { <. <. x , y >. , z >. | ( x e. A /\ y e. B /\ z e. C ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z e. C ) }
6	2 3 5	3eqtr4i	|- ( ( A X. B ) X. C ) = { <. <. x , y >. , z >. | ( x e. A /\ y e. B /\ z e. C ) }