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Metamath Proof Explorer

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 ) }