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Theorem dmxpm 4729
Description: The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmxpm  |-  ( E. x  x  e.  B  ->  dom  ( A  X.  B )  =  A )
Distinct variable group:    x, B
Allowed substitution hint:    A( x)

Proof of Theorem dmxpm
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2180 . . 3  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
21cbvexv 1872 . 2  |-  ( E. x  x  e.  B  <->  E. z  z  e.  B
)
3 df-xp 4515 . . . 4  |-  ( A  X.  B )  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
43dmeqi 4710 . . 3  |-  dom  ( A  X.  B )  =  dom  { <. y ,  z >.  |  ( y  e.  A  /\  z  e.  B ) }
5 id 19 . . . . 5  |-  ( E. z  z  e.  B  ->  E. z  z  e.  B )
65ralrimivw 2483 . . . 4  |-  ( E. z  z  e.  B  ->  A. y  e.  A  E. z  z  e.  B )
7 dmopab3 4722 . . . 4  |-  ( A. y  e.  A  E. z  z  e.  B  <->  dom 
{ <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }  =  A )
86, 7sylib 121 . . 3  |-  ( E. z  z  e.  B  ->  dom  { <. y ,  z >.  |  ( y  e.  A  /\  z  e.  B ) }  =  A )
94, 8syl5eq 2162 . 2  |-  ( E. z  z  e.  B  ->  dom  ( A  X.  B )  =  A )
102, 9sylbi 120 1  |-  ( E. x  x  e.  B  ->  dom  ( A  X.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   A.wral 2393   {copab 3958    X. cxp 4507   dom cdm 4509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-dm 4519
This theorem is referenced by:  rnxpm  4938  ssxpbm  4944  ssxp1  4945  xpexr2m  4950  relrelss  5035  unixpm  5044  exmidfodomrlemim  7025
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