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Metamath Proof Explorer

Theorem inxp

Description: Intersection of two Cartesian products. Exercise 9 of TakeutiZaring p. 25. (Contributed by NM, 3-Aug-1994) (Proof shortened by Andrew Salmon, 27-Aug-2011) Avoid ax-10 , ax-12 . (Revised by SN, 5-May-2025)

Ref Expression
Assertion inxp 
|- ( ( A X. B ) i^i ( C X. D ) ) = ( ( A i^i C ) X. ( B i^i D ) )

Proof

Step Hyp Ref Expression
1 relinxp 
 |-  Rel ( ( A X. B ) i^i ( C X. D ) )
2 relxp 
 |-  Rel ( ( A i^i C ) X. ( B i^i D ) )
3 an4 
 |-  ( ( ( x e. A /\ y e. B ) /\ ( x e. C /\ y e. D ) ) <-> ( ( x e. A /\ x e. C ) /\ ( y e. B /\ y e. D ) ) )
4 opelxp 
 |-  ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
5 opelxp 
 |-  ( <. x , y >. e. ( C X. D ) <-> ( x e. C /\ y e. D ) )
6 4 5 anbi12i 
 |-  ( ( <. x , y >. e. ( A X. B ) /\ <. x , y >. e. ( C X. D ) ) <-> ( ( x e. A /\ y e. B ) /\ ( x e. C /\ y e. D ) ) )
7 elin 
 |-  ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
8 elin 
 |-  ( y e. ( B i^i D ) <-> ( y e. B /\ y e. D ) )
9 7 8 anbi12i 
 |-  ( ( x e. ( A i^i C ) /\ y e. ( B i^i D ) ) <-> ( ( x e. A /\ x e. C ) /\ ( y e. B /\ y e. D ) ) )
10 3 6 9 3bitr4i 
 |-  ( ( <. x , y >. e. ( A X. B ) /\ <. x , y >. e. ( C X. D ) ) <-> ( x e. ( A i^i C ) /\ y e. ( B i^i D ) ) )
11 elin 
 |-  ( <. x , y >. e. ( ( A X. B ) i^i ( C X. D ) ) <-> ( <. x , y >. e. ( A X. B ) /\ <. x , y >. e. ( C X. D ) ) )
12 opelxp 
 |-  ( <. x , y >. e. ( ( A i^i C ) X. ( B i^i D ) ) <-> ( x e. ( A i^i C ) /\ y e. ( B i^i D ) ) )
13 10 11 12 3bitr4i 
 |-  ( <. x , y >. e. ( ( A X. B ) i^i ( C X. D ) ) <-> <. x , y >. e. ( ( A i^i C ) X. ( B i^i D ) ) )
14 1 2 13 eqrelriiv 
 |-  ( ( A X. B ) i^i ( C X. D ) ) = ( ( A i^i C ) X. ( B i^i D ) )