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Theorem dmxpin 5091
Description: The domain of the intersection of two square cross products. Unlike dmin 5078, equality holds. (Contributed by NM, 29-Jan-2008.)
Assertion
Ref Expression
dmxpin  |-  dom  (
( A  X.  A
)  i^i  ( B  X.  B ) )  =  ( A  i^i  B
)

Proof of Theorem dmxpin
StepHypRef Expression
1 inxp 5008 . . 3  |-  ( ( A  X.  A )  i^i  ( B  X.  B ) )  =  ( ( A  i^i  B )  X.  ( A  i^i  B ) )
21dmeqi 5072 . 2  |-  dom  (
( A  X.  A
)  i^i  ( B  X.  B ) )  =  dom  ( ( A  i^i  B )  X.  ( A  i^i  B
) )
3 dmxpid 5090 . 2  |-  dom  (
( A  i^i  B
)  X.  ( A  i^i  B ) )  =  ( A  i^i  B )
42, 3eqtri 2457 1  |-  dom  (
( A  X.  A
)  i^i  ( B  X.  B ) )  =  ( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    i^i cin 3320    X. cxp 4877   dom cdm 4879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-dm 4889
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