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Theorem dmxpm 4583
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 2142 . . 3  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
21cbvexv 1837 . 2  |-  ( E. x  x  e.  B  <->  E. z  z  e.  B
)
3 df-xp 4377 . . . 4  |-  ( A  X.  B )  =  { <. y ,  z
>.  |  ( y  e.  A  /\  z  e.  B ) }
43dmeqi 4564 . . 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 2436 . . . 4  |-  ( E. z  z  e.  B  ->  A. y  e.  A  E. z  z  e.  B )
7 dmopab3 4576 . . . 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 2126 . 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 1285   E.wex 1422    e. wcel 1434   A.wral 2349   {copab 3846    X. cxp 4369   dom cdm 4371
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-dm 4381
This theorem is referenced by:  dmxpinm  4584  xpid11m  4585  rnxpm  4782  ssxpbm  4786  ssxp1  4787  xpexr2m  4792  relrelss  4874  unixpm  4883  xpiderm  6243
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