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Theorem dmxpinm 4556
Description: The domain of the intersection of two square cross products. Unlike dmin 4543, equality holds. (Contributed by NM, 29-Jan-2008.)
Assertion
Ref Expression
dmxpinm  |-  ( E. x  x  e.  ( A  i^i  B )  ->  dom  ( ( A  X.  A )  i^i  ( B  X.  B
) )  =  ( A  i^i  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem dmxpinm
StepHypRef Expression
1 inxp 4470 . . . 4  |-  ( ( A  X.  A )  i^i  ( B  X.  B ) )  =  ( ( A  i^i  B )  X.  ( A  i^i  B ) )
21dmeqi 4536 . . 3  |-  dom  (
( A  X.  A
)  i^i  ( B  X.  B ) )  =  dom  ( ( A  i^i  B )  X.  ( A  i^i  B
) )
32a1i 9 . 2  |-  ( E. x  x  e.  ( A  i^i  B )  ->  dom  ( ( A  X.  A )  i^i  ( B  X.  B
) )  =  dom  ( ( A  i^i  B )  X.  ( A  i^i  B ) ) )
4 dmxpm 4555 . 2  |-  ( E. x  x  e.  ( A  i^i  B )  ->  dom  ( ( A  i^i  B )  X.  ( A  i^i  B
) )  =  ( A  i^i  B ) )
53, 4eqtrd 2072 1  |-  ( E. x  x  e.  ( A  i^i  B )  ->  dom  ( ( A  X.  A )  i^i  ( B  X.  B
) )  =  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   E.wex 1381    e. wcel 1393    i^i cin 2916    X. cxp 4343   dom cdm 4345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355
This theorem is referenced by: (None)
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