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Theorem dmxpm 4644
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 2150 . . 3  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
21cbvexv 1843 . 2  |-  ( E. x  x  e.  B  <->  E. z  z  e.  B
)
3 df-xp 4434 . . . 4  |-  ( A  X.  B )  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
43dmeqi 4625 . . 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 2447 . . . 4  |-  ( E. z  z  e.  B  ->  A. y  e.  A  E. z  z  e.  B )
7 dmopab3 4637 . . . 4  |-  ( A. y  e.  A  E. z  z  e.  B  <->  dom 
{ <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }  =  A )
86, 7sylib 120 . . 3  |-  ( E. z  z  e.  B  ->  dom  { <. y ,  z >.  |  ( y  e.  A  /\  z  e.  B ) }  =  A )
94, 8syl5eq 2132 . 2  |-  ( E. z  z  e.  B  ->  dom  ( A  X.  B )  =  A )
102, 9sylbi 119 1  |-  ( E. x  x  e.  B  ->  dom  ( A  X.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   A.wral 2359   {copab 3890    X. cxp 4426   dom cdm 4428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-dm 4438
This theorem is referenced by:  dmxpinm  4645  xpid11m  4646  rnxpm  4847  ssxpbm  4853  ssxp1  4854  xpexr2m  4859  relrelss  4944  unixpm  4953  xpiderm  6343  exmidfodomrlemim  6806
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